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Properties

Label 24-3e72-1.1-c3e12-0-0
Degree 2424
Conductor 2.253×10342.253\times 10^{34}
Sign 11
Analytic cond. 4.00980×10194.00980\times 10^{19}
Root an. cond. 6.558386.55838
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 1212

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 12·4-s − 12·5-s − 42·7-s + 153·8-s + 72·10-s − 42·11-s − 78·13-s + 252·14-s − 150·16-s + 18·17-s − 228·19-s + 144·20-s + 252·22-s + 114·23-s − 687·25-s + 468·26-s + 504·28-s + 660·29-s − 708·31-s − 1.54e3·32-s − 108·34-s + 504·35-s − 354·37-s + 1.36e3·38-s − 1.83e3·40-s + 1.03e3·41-s + ⋯
L(s)  = 1  − 2.12·2-s − 3/2·4-s − 1.07·5-s − 2.26·7-s + 6.76·8-s + 2.27·10-s − 1.15·11-s − 1.66·13-s + 4.81·14-s − 2.34·16-s + 0.256·17-s − 2.75·19-s + 1.60·20-s + 2.44·22-s + 1.03·23-s − 5.49·25-s + 3.53·26-s + 3.40·28-s + 4.22·29-s − 4.10·31-s − 8.53·32-s − 0.544·34-s + 2.43·35-s − 1.57·37-s + 5.83·38-s − 7.25·40-s + 3.93·41-s + ⋯

Functional equation

Λ(s)=((372)s/2 ΓC(s)12 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((372)s/2 ΓC(s+3/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 3723^{72}
Sign: 11
Analytic conductor: 4.00980×10194.00980\times 10^{19}
Root analytic conductor: 6.558386.55838
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 1212
Selberg data: (24, 372, ( :[3/2]12), 1)(24,\ 3^{72} ,\ ( \ : [3/2]^{12} ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
good2 1+3pT+3p4T2+207T3+525pT4+3885T5+8077pT6+55377T7+201177T8+635877T9+1026261pT10+747189p3T11+2205423p3T12+747189p6T13+1026261p7T14+635877p9T15+201177p12T16+55377p15T17+8077p19T18+3885p21T19+525p25T20+207p27T21+3p34T22+3p34T23+p36T24 1 + 3 p T + 3 p^{4} T^{2} + 207 T^{3} + 525 p T^{4} + 3885 T^{5} + 8077 p T^{6} + 55377 T^{7} + 201177 T^{8} + 635877 T^{9} + 1026261 p T^{10} + 747189 p^{3} T^{11} + 2205423 p^{3} T^{12} + 747189 p^{6} T^{13} + 1026261 p^{7} T^{14} + 635877 p^{9} T^{15} + 201177 p^{12} T^{16} + 55377 p^{15} T^{17} + 8077 p^{19} T^{18} + 3885 p^{21} T^{19} + 525 p^{25} T^{20} + 207 p^{27} T^{21} + 3 p^{34} T^{22} + 3 p^{34} T^{23} + p^{36} T^{24}
5 1+12T+831T2+18p4T3+369699T4+5062422T5+113589566T6+1473568758T7+26132727888T8+310624976034T9+4660139652948T10+49912371950148T11+654659995378509T12+49912371950148p3T13+4660139652948p6T14+310624976034p9T15+26132727888p12T16+1473568758p15T17+113589566p18T18+5062422p21T19+369699p24T20+18p31T21+831p30T22+12p33T23+p36T24 1 + 12 T + 831 T^{2} + 18 p^{4} T^{3} + 369699 T^{4} + 5062422 T^{5} + 113589566 T^{6} + 1473568758 T^{7} + 26132727888 T^{8} + 310624976034 T^{9} + 4660139652948 T^{10} + 49912371950148 T^{11} + 654659995378509 T^{12} + 49912371950148 p^{3} T^{13} + 4660139652948 p^{6} T^{14} + 310624976034 p^{9} T^{15} + 26132727888 p^{12} T^{16} + 1473568758 p^{15} T^{17} + 113589566 p^{18} T^{18} + 5062422 p^{21} T^{19} + 369699 p^{24} T^{20} + 18 p^{31} T^{21} + 831 p^{30} T^{22} + 12 p^{33} T^{23} + p^{36} T^{24}
7 1+6pT+3036T2+101842T3+615315pT4+121274466T5+3874201291T6+94568685912T7+2503702606005T8+53969769936898T9+176352850122207pT10+23664619067852244T11+476742898558106410T12+23664619067852244p3T13+176352850122207p7T14+53969769936898p9T15+2503702606005p12T16+94568685912p15T17+3874201291p18T18+121274466p21T19+615315p25T20+101842p27T21+3036p30T22+6p34T23+p36T24 1 + 6 p T + 3036 T^{2} + 101842 T^{3} + 615315 p T^{4} + 121274466 T^{5} + 3874201291 T^{6} + 94568685912 T^{7} + 2503702606005 T^{8} + 53969769936898 T^{9} + 176352850122207 p T^{10} + 23664619067852244 T^{11} + 476742898558106410 T^{12} + 23664619067852244 p^{3} T^{13} + 176352850122207 p^{7} T^{14} + 53969769936898 p^{9} T^{15} + 2503702606005 p^{12} T^{16} + 94568685912 p^{15} T^{17} + 3874201291 p^{18} T^{18} + 121274466 p^{21} T^{19} + 615315 p^{25} T^{20} + 101842 p^{27} T^{21} + 3036 p^{30} T^{22} + 6 p^{34} T^{23} + p^{36} T^{24}
11 1+42T+10938T2+399708T3+57060717T4+1846526892T5+190126522637T6+5522021579286T7+457335567708237T8+12021670470373236T9+848270152906356753T10+20203589745149563296T11+ 1 + 42 T + 10938 T^{2} + 399708 T^{3} + 57060717 T^{4} + 1846526892 T^{5} + 190126522637 T^{6} + 5522021579286 T^{7} + 457335567708237 T^{8} + 12021670470373236 T^{9} + 848270152906356753 T^{10} + 20203589745149563296 T^{11} + 12 ⁣⋯ ⁣1812\!\cdots\!18T12+20203589745149563296p3T13+848270152906356753p6T14+12021670470373236p9T15+457335567708237p12T16+5522021579286p15T17+190126522637p18T18+1846526892p21T19+57060717p24T20+399708p27T21+10938p30T22+42p33T23+p36T24 T^{12} + 20203589745149563296 p^{3} T^{13} + 848270152906356753 p^{6} T^{14} + 12021670470373236 p^{9} T^{15} + 457335567708237 p^{12} T^{16} + 5522021579286 p^{15} T^{17} + 190126522637 p^{18} T^{18} + 1846526892 p^{21} T^{19} + 57060717 p^{24} T^{20} + 399708 p^{27} T^{21} + 10938 p^{30} T^{22} + 42 p^{33} T^{23} + p^{36} T^{24}
13 1+6pT+17598T2+1171762T3+144122232T4+8131311312T5+727323282172T6+35054502969726T7+2575226035123182T8+108478974509515420T9+7080074634468856686T10+ 1 + 6 p T + 17598 T^{2} + 1171762 T^{3} + 144122232 T^{4} + 8131311312 T^{5} + 727323282172 T^{6} + 35054502969726 T^{7} + 2575226035123182 T^{8} + 108478974509515420 T^{9} + 7080074634468856686 T^{10} + 27 ⁣⋯ ⁣5827\!\cdots\!58T11+ T^{11} + 16 ⁣⋯ ⁣8116\!\cdots\!81T12+ T^{12} + 27 ⁣⋯ ⁣5827\!\cdots\!58p3T13+7080074634468856686p6T14+108478974509515420p9T15+2575226035123182p12T16+35054502969726p15T17+727323282172p18T18+8131311312p21T19+144122232p24T20+1171762p27T21+17598p30T22+6p34T23+p36T24 p^{3} T^{13} + 7080074634468856686 p^{6} T^{14} + 108478974509515420 p^{9} T^{15} + 2575226035123182 p^{12} T^{16} + 35054502969726 p^{15} T^{17} + 727323282172 p^{18} T^{18} + 8131311312 p^{21} T^{19} + 144122232 p^{24} T^{20} + 1171762 p^{27} T^{21} + 17598 p^{30} T^{22} + 6 p^{34} T^{23} + p^{36} T^{24}
17 1−18T+1605pT2−46188T3+378799590T4+3970670040T5+3687982052216T6+68480154333612T7+28334851651381545T8+642470064566247468T9+ 1 - 18 T + 1605 p T^{2} - 46188 T^{3} + 378799590 T^{4} + 3970670040 T^{5} + 3687982052216 T^{6} + 68480154333612 T^{7} + 28334851651381545 T^{8} + 642470064566247468 T^{9} + 17 ⁣⋯ ⁣2317\!\cdots\!23T10+ T^{10} + 42 ⁣⋯ ⁣0642\!\cdots\!06T11+ T^{11} + 56 ⁣⋯ ⁣0156\!\cdots\!01pT12+ p T^{12} + 42 ⁣⋯ ⁣0642\!\cdots\!06p3T13+ p^{3} T^{13} + 17 ⁣⋯ ⁣2317\!\cdots\!23p6T14+642470064566247468p9T15+28334851651381545p12T16+68480154333612p15T17+3687982052216p18T18+3970670040p21T19+378799590p24T20−46188p27T21+1605p31T22−18p33T23+p36T24 p^{6} T^{14} + 642470064566247468 p^{9} T^{15} + 28334851651381545 p^{12} T^{16} + 68480154333612 p^{15} T^{17} + 3687982052216 p^{18} T^{18} + 3970670040 p^{21} T^{19} + 378799590 p^{24} T^{20} - 46188 p^{27} T^{21} + 1605 p^{31} T^{22} - 18 p^{33} T^{23} + p^{36} T^{24}
19 1+12pT+65598T2+10557550T3+1895939457T4+244909953084T5+34002829867213T6+3727551684296016T7+433911156710034501T8+41620379928293229856T9+ 1 + 12 p T + 65598 T^{2} + 10557550 T^{3} + 1895939457 T^{4} + 244909953084 T^{5} + 34002829867213 T^{6} + 3727551684296016 T^{7} + 433911156710034501 T^{8} + 41620379928293229856 T^{9} + 42 ⁣⋯ ⁣6142\!\cdots\!61T10+ T^{10} + 18 ⁣⋯ ⁣2618\!\cdots\!26pT11+ p T^{11} + 32 ⁣⋯ ⁣5032\!\cdots\!50T12+ T^{12} + 18 ⁣⋯ ⁣2618\!\cdots\!26p4T13+ p^{4} T^{13} + 42 ⁣⋯ ⁣6142\!\cdots\!61p6T14+41620379928293229856p9T15+433911156710034501p12T16+3727551684296016p15T17+34002829867213p18T18+244909953084p21T19+1895939457p24T20+10557550p27T21+65598p30T22+12p34T23+p36T24 p^{6} T^{14} + 41620379928293229856 p^{9} T^{15} + 433911156710034501 p^{12} T^{16} + 3727551684296016 p^{15} T^{17} + 34002829867213 p^{18} T^{18} + 244909953084 p^{21} T^{19} + 1895939457 p^{24} T^{20} + 10557550 p^{27} T^{21} + 65598 p^{30} T^{22} + 12 p^{34} T^{23} + p^{36} T^{24}
23 1−114T+86844T2−7473096T3+3589570059T4−9934897818pT5+94603846309235T6−4216862110588992T7+1815581284007638833T8−53177452533524291454T9+ 1 - 114 T + 86844 T^{2} - 7473096 T^{3} + 3589570059 T^{4} - 9934897818 p T^{5} + 94603846309235 T^{6} - 4216862110588992 T^{7} + 1815581284007638833 T^{8} - 53177452533524291454 T^{9} + 27 ⁣⋯ ⁣9727\!\cdots\!97T10− T^{10} - 55 ⁣⋯ ⁣0655\!\cdots\!06T11+ T^{11} + 36 ⁣⋯ ⁣9836\!\cdots\!98T12− T^{12} - 55 ⁣⋯ ⁣0655\!\cdots\!06p3T13+ p^{3} T^{13} + 27 ⁣⋯ ⁣9727\!\cdots\!97p6T14−53177452533524291454p9T15+1815581284007638833p12T16−4216862110588992p15T17+94603846309235p18T18−9934897818p22T19+3589570059p24T20−7473096p27T21+86844p30T22−114p33T23+p36T24 p^{6} T^{14} - 53177452533524291454 p^{9} T^{15} + 1815581284007638833 p^{12} T^{16} - 4216862110588992 p^{15} T^{17} + 94603846309235 p^{18} T^{18} - 9934897818 p^{22} T^{19} + 3589570059 p^{24} T^{20} - 7473096 p^{27} T^{21} + 86844 p^{30} T^{22} - 114 p^{33} T^{23} + p^{36} T^{24}
29 1−660T+333813T2−118267254T3+36609873711T4−9440888603268T5+2231799317094842T6−466678842948720126T7+92048488913551542102T8− 1 - 660 T + 333813 T^{2} - 118267254 T^{3} + 36609873711 T^{4} - 9440888603268 T^{5} + 2231799317094842 T^{6} - 466678842948720126 T^{7} + 92048488913551542102 T^{8} - 16 ⁣⋯ ⁣5416\!\cdots\!54T9+ T^{9} + 28 ⁣⋯ ⁣8828\!\cdots\!88T10− T^{10} - 47 ⁣⋯ ⁣5247\!\cdots\!52T11+ T^{11} + 75 ⁣⋯ ⁣5775\!\cdots\!57T12− T^{12} - 47 ⁣⋯ ⁣5247\!\cdots\!52p3T13+ p^{3} T^{13} + 28 ⁣⋯ ⁣8828\!\cdots\!88p6T14− p^{6} T^{14} - 16 ⁣⋯ ⁣5416\!\cdots\!54p9T15+92048488913551542102p12T16−466678842948720126p15T17+2231799317094842p18T18−9440888603268p21T19+36609873711p24T20−118267254p27T21+333813p30T22−660p33T23+p36T24 p^{9} T^{15} + 92048488913551542102 p^{12} T^{16} - 466678842948720126 p^{15} T^{17} + 2231799317094842 p^{18} T^{18} - 9440888603268 p^{21} T^{19} + 36609873711 p^{24} T^{20} - 118267254 p^{27} T^{21} + 333813 p^{30} T^{22} - 660 p^{33} T^{23} + p^{36} T^{24}
31 1+708T+396237T2+152780596T3+52458006351T4+15058806235008T5+4064767979251492T6+980515278323793720T7+ 1 + 708 T + 396237 T^{2} + 152780596 T^{3} + 52458006351 T^{4} + 15058806235008 T^{5} + 4064767979251492 T^{6} + 980515278323793720 T^{7} + 22 ⁣⋯ ⁣5722\!\cdots\!57T8+ T^{8} + 47 ⁣⋯ ⁣5247\!\cdots\!52T9+ T^{9} + 96 ⁣⋯ ⁣1196\!\cdots\!11T10+ T^{10} + 17 ⁣⋯ ⁣6817\!\cdots\!68T11+ T^{11} + 32 ⁣⋯ ⁣7032\!\cdots\!70T12+ T^{12} + 17 ⁣⋯ ⁣6817\!\cdots\!68p3T13+ p^{3} T^{13} + 96 ⁣⋯ ⁣1196\!\cdots\!11p6T14+ p^{6} T^{14} + 47 ⁣⋯ ⁣5247\!\cdots\!52p9T15+ p^{9} T^{15} + 22 ⁣⋯ ⁣5722\!\cdots\!57p12T16+980515278323793720p15T17+4064767979251492p18T18+15058806235008p21T19+52458006351p24T20+152780596p27T21+396237p30T22+708p33T23+p36T24 p^{12} T^{16} + 980515278323793720 p^{15} T^{17} + 4064767979251492 p^{18} T^{18} + 15058806235008 p^{21} T^{19} + 52458006351 p^{24} T^{20} + 152780596 p^{27} T^{21} + 396237 p^{30} T^{22} + 708 p^{33} T^{23} + p^{36} T^{24}
37 1+354T+241377T2+74455912T3+34606095729T4+9686749492272T5+3561065782685848T6+924024438254694024T7+ 1 + 354 T + 241377 T^{2} + 74455912 T^{3} + 34606095729 T^{4} + 9686749492272 T^{5} + 3561065782685848 T^{6} + 924024438254694024 T^{7} + 29 ⁣⋯ ⁣7029\!\cdots\!70T8+ T^{8} + 69 ⁣⋯ ⁣2069\!\cdots\!20T9+ T^{9} + 19 ⁣⋯ ⁣6419\!\cdots\!64T10+ T^{10} + 42 ⁣⋯ ⁣8442\!\cdots\!84T11+ T^{11} + 10 ⁣⋯ ⁣8110\!\cdots\!81T12+ T^{12} + 42 ⁣⋯ ⁣8442\!\cdots\!84p3T13+ p^{3} T^{13} + 19 ⁣⋯ ⁣6419\!\cdots\!64p6T14+ p^{6} T^{14} + 69 ⁣⋯ ⁣2069\!\cdots\!20p9T15+ p^{9} T^{15} + 29 ⁣⋯ ⁣7029\!\cdots\!70p12T16+924024438254694024p15T17+3561065782685848p18T18+9686749492272p21T19+34606095729p24T20+74455912p27T21+241377p30T22+354p33T23+p36T24 p^{12} T^{16} + 924024438254694024 p^{15} T^{17} + 3561065782685848 p^{18} T^{18} + 9686749492272 p^{21} T^{19} + 34606095729 p^{24} T^{20} + 74455912 p^{27} T^{21} + 241377 p^{30} T^{22} + 354 p^{33} T^{23} + p^{36} T^{24}
41 1−1032T+921684T2−562850946T3+310655218110T4−142597486514472T5+60722193909672899T6−22922842456199835774T7+ 1 - 1032 T + 921684 T^{2} - 562850946 T^{3} + 310655218110 T^{4} - 142597486514472 T^{5} + 60722193909672899 T^{6} - 22922842456199835774 T^{7} + 81 ⁣⋯ ⁣3581\!\cdots\!35T8− T^{8} - 26 ⁣⋯ ⁣2426\!\cdots\!24T9+ T^{9} + 81 ⁣⋯ ⁣6281\!\cdots\!62T10− T^{10} - 23 ⁣⋯ ⁣3423\!\cdots\!34T11+ T^{11} + 63 ⁣⋯ ⁣9163\!\cdots\!91T12− T^{12} - 23 ⁣⋯ ⁣3423\!\cdots\!34p3T13+ p^{3} T^{13} + 81 ⁣⋯ ⁣6281\!\cdots\!62p6T14− p^{6} T^{14} - 26 ⁣⋯ ⁣2426\!\cdots\!24p9T15+ p^{9} T^{15} + 81 ⁣⋯ ⁣3581\!\cdots\!35p12T16−22922842456199835774p15T17+60722193909672899p18T18−142597486514472p21T19+310655218110p24T20−562850946p27T21+921684p30T22−1032p33T23+p36T24 p^{12} T^{16} - 22922842456199835774 p^{15} T^{17} + 60722193909672899 p^{18} T^{18} - 142597486514472 p^{21} T^{19} + 310655218110 p^{24} T^{20} - 562850946 p^{27} T^{21} + 921684 p^{30} T^{22} - 1032 p^{33} T^{23} + p^{36} T^{24}
43 1+744T+553692T2+225937744T3+104527998066T4+33437243209776T5+14025330871137100T6+4278437402978673432T7+ 1 + 744 T + 553692 T^{2} + 225937744 T^{3} + 104527998066 T^{4} + 33437243209776 T^{5} + 14025330871137100 T^{6} + 4278437402978673432 T^{7} + 16 ⁣⋯ ⁣2716\!\cdots\!27T8+ T^{8} + 47 ⁣⋯ ⁣8047\!\cdots\!80T9+ T^{9} + 16 ⁣⋯ ⁣6816\!\cdots\!68T10+ T^{10} + 42 ⁣⋯ ⁣1642\!\cdots\!16T11+ T^{11} + 14 ⁣⋯ ⁣9214\!\cdots\!92T12+ T^{12} + 42 ⁣⋯ ⁣1642\!\cdots\!16p3T13+ p^{3} T^{13} + 16 ⁣⋯ ⁣6816\!\cdots\!68p6T14+ p^{6} T^{14} + 47 ⁣⋯ ⁣8047\!\cdots\!80p9T15+ p^{9} T^{15} + 16 ⁣⋯ ⁣2716\!\cdots\!27p12T16+4278437402978673432p15T17+14025330871137100p18T18+33437243209776p21T19+104527998066p24T20+225937744p27T21+553692p30T22+744p33T23+p36T24 p^{12} T^{16} + 4278437402978673432 p^{15} T^{17} + 14025330871137100 p^{18} T^{18} + 33437243209776 p^{21} T^{19} + 104527998066 p^{24} T^{20} + 225937744 p^{27} T^{21} + 553692 p^{30} T^{22} + 744 p^{33} T^{23} + p^{36} T^{24}
47 1+942T+1058889T2+690880410T3+471117013206T4+241228679983080T5+125419907597914160T6+53682080815284171714T7+ 1 + 942 T + 1058889 T^{2} + 690880410 T^{3} + 471117013206 T^{4} + 241228679983080 T^{5} + 125419907597914160 T^{6} + 53682080815284171714 T^{7} + 23 ⁣⋯ ⁣1323\!\cdots\!13T8+ T^{8} + 86 ⁣⋯ ⁣4486\!\cdots\!44T9+ T^{9} + 32 ⁣⋯ ⁣7132\!\cdots\!71T10+ T^{10} + 10 ⁣⋯ ⁣4610\!\cdots\!46T11+ T^{11} + 37 ⁣⋯ ⁣6037\!\cdots\!60T12+ T^{12} + 10 ⁣⋯ ⁣4610\!\cdots\!46p3T13+ p^{3} T^{13} + 32 ⁣⋯ ⁣7132\!\cdots\!71p6T14+ p^{6} T^{14} + 86 ⁣⋯ ⁣4486\!\cdots\!44p9T15+ p^{9} T^{15} + 23 ⁣⋯ ⁣1323\!\cdots\!13p12T16+53682080815284171714p15T17+125419907597914160p18T18+241228679983080p21T19+471117013206p24T20+690880410p27T21+1058889p30T22+942p33T23+p36T24 p^{12} T^{16} + 53682080815284171714 p^{15} T^{17} + 125419907597914160 p^{18} T^{18} + 241228679983080 p^{21} T^{19} + 471117013206 p^{24} T^{20} + 690880410 p^{27} T^{21} + 1058889 p^{30} T^{22} + 942 p^{33} T^{23} + p^{36} T^{24}
53 1−828T+1410657T2−990164700T3+926325130575T4−563788608990552T5+379915391732165060T6− 1 - 828 T + 1410657 T^{2} - 990164700 T^{3} + 926325130575 T^{4} - 563788608990552 T^{5} + 379915391732165060 T^{6} - 20 ⁣⋯ ⁣5220\!\cdots\!52T7+ T^{7} + 10 ⁣⋯ ⁣8510\!\cdots\!85T8− T^{8} - 52 ⁣⋯ ⁣1652\!\cdots\!16T9+ T^{9} + 23 ⁣⋯ ⁣5923\!\cdots\!59T10− T^{10} - 10 ⁣⋯ ⁣9210\!\cdots\!92T11+ T^{11} + 40 ⁣⋯ ⁣5840\!\cdots\!58T12− T^{12} - 10 ⁣⋯ ⁣9210\!\cdots\!92p3T13+ p^{3} T^{13} + 23 ⁣⋯ ⁣5923\!\cdots\!59p6T14− p^{6} T^{14} - 52 ⁣⋯ ⁣1652\!\cdots\!16p9T15+ p^{9} T^{15} + 10 ⁣⋯ ⁣8510\!\cdots\!85p12T16− p^{12} T^{16} - 20 ⁣⋯ ⁣5220\!\cdots\!52p15T17+379915391732165060p18T18−563788608990552p21T19+926325130575p24T20−990164700p27T21+1410657p30T22−828p33T23+p36T24 p^{15} T^{17} + 379915391732165060 p^{18} T^{18} - 563788608990552 p^{21} T^{19} + 926325130575 p^{24} T^{20} - 990164700 p^{27} T^{21} + 1410657 p^{30} T^{22} - 828 p^{33} T^{23} + p^{36} T^{24}
59 1−24T+1154712T2+108879264T3+654469055961T4+142630153314726T5+256843348960078295T6+79288323473587318434T7+ 1 - 24 T + 1154712 T^{2} + 108879264 T^{3} + 654469055961 T^{4} + 142630153314726 T^{5} + 256843348960078295 T^{6} + 79288323473587318434 T^{7} + 82 ⁣⋯ ⁣2582\!\cdots\!25T8+ T^{8} + 27 ⁣⋯ ⁣0627\!\cdots\!06T9+ T^{9} + 22 ⁣⋯ ⁣0522\!\cdots\!05T10+ T^{10} + 69 ⁣⋯ ⁣9469\!\cdots\!94T11+ T^{11} + 51 ⁣⋯ ⁣1851\!\cdots\!18T12+ T^{12} + 69 ⁣⋯ ⁣9469\!\cdots\!94p3T13+ p^{3} T^{13} + 22 ⁣⋯ ⁣0522\!\cdots\!05p6T14+ p^{6} T^{14} + 27 ⁣⋯ ⁣0627\!\cdots\!06p9T15+ p^{9} T^{15} + 82 ⁣⋯ ⁣2582\!\cdots\!25p12T16+79288323473587318434p15T17+256843348960078295p18T18+142630153314726p21T19+654469055961p24T20+108879264p27T21+1154712p30T22−24p33T23+p36T24 p^{12} T^{16} + 79288323473587318434 p^{15} T^{17} + 256843348960078295 p^{18} T^{18} + 142630153314726 p^{21} T^{19} + 654469055961 p^{24} T^{20} + 108879264 p^{27} T^{21} + 1154712 p^{30} T^{22} - 24 p^{33} T^{23} + p^{36} T^{24}
61 1+1698T+2661744T2+2733019648T3+2588156484195T4+1954552764603318T5+1384169662080875359T6+ 1 + 1698 T + 2661744 T^{2} + 2733019648 T^{3} + 2588156484195 T^{4} + 1954552764603318 T^{5} + 1384169662080875359 T^{6} + 83 ⁣⋯ ⁣1683\!\cdots\!16T7+ T^{7} + 48 ⁣⋯ ⁣8948\!\cdots\!89T8+ T^{8} + 24 ⁣⋯ ⁣5424\!\cdots\!54T9+ T^{9} + 12 ⁣⋯ ⁣9712\!\cdots\!97T10+ T^{10} + 59 ⁣⋯ ⁣1059\!\cdots\!10T11+ T^{11} + 28 ⁣⋯ ⁣8628\!\cdots\!86T12+ T^{12} + 59 ⁣⋯ ⁣1059\!\cdots\!10p3T13+ p^{3} T^{13} + 12 ⁣⋯ ⁣9712\!\cdots\!97p6T14+ p^{6} T^{14} + 24 ⁣⋯ ⁣5424\!\cdots\!54p9T15+ p^{9} T^{15} + 48 ⁣⋯ ⁣8948\!\cdots\!89p12T16+ p^{12} T^{16} + 83 ⁣⋯ ⁣1683\!\cdots\!16p15T17+1384169662080875359p18T18+1954552764603318p21T19+2588156484195p24T20+2733019648p27T21+2661744p30T22+1698p33T23+p36T24 p^{15} T^{17} + 1384169662080875359 p^{18} T^{18} + 1954552764603318 p^{21} T^{19} + 2588156484195 p^{24} T^{20} + 2733019648 p^{27} T^{21} + 2661744 p^{30} T^{22} + 1698 p^{33} T^{23} + p^{36} T^{24}
67 1+1266T+2242281T2+2079332890T3+2203397904414T4+1667100797153556T5+1356716020460611204T6+ 1 + 1266 T + 2242281 T^{2} + 2079332890 T^{3} + 2203397904414 T^{4} + 1667100797153556 T^{5} + 1356716020460611204 T^{6} + 89 ⁣⋯ ⁣9889\!\cdots\!98T7+ T^{7} + 61 ⁣⋯ ⁣0561\!\cdots\!05T8+ T^{8} + 36 ⁣⋯ ⁣4836\!\cdots\!48T9+ T^{9} + 22 ⁣⋯ ⁣2722\!\cdots\!27T10+ T^{10} + 12 ⁣⋯ ⁣7812\!\cdots\!78T11+ T^{11} + 72 ⁣⋯ ⁣6072\!\cdots\!60T12+ T^{12} + 12 ⁣⋯ ⁣7812\!\cdots\!78p3T13+ p^{3} T^{13} + 22 ⁣⋯ ⁣2722\!\cdots\!27p6T14+ p^{6} T^{14} + 36 ⁣⋯ ⁣4836\!\cdots\!48p9T15+ p^{9} T^{15} + 61 ⁣⋯ ⁣0561\!\cdots\!05p12T16+ p^{12} T^{16} + 89 ⁣⋯ ⁣9889\!\cdots\!98p15T17+1356716020460611204p18T18+1667100797153556p21T19+2203397904414p24T20+2079332890p27T21+2242281p30T22+1266p33T23+p36T24 p^{15} T^{17} + 1356716020460611204 p^{18} T^{18} + 1667100797153556 p^{21} T^{19} + 2203397904414 p^{24} T^{20} + 2079332890 p^{27} T^{21} + 2242281 p^{30} T^{22} + 1266 p^{33} T^{23} + p^{36} T^{24}
71 1+3888T+9472848T2+16698964968T3+23909532632490T4+28876815994589208T5+30580771234603876304T6+ 1 + 3888 T + 9472848 T^{2} + 16698964968 T^{3} + 23909532632490 T^{4} + 28876815994589208 T^{5} + 30580771234603876304 T^{6} + 28 ⁣⋯ ⁣9228\!\cdots\!92T7+ T^{7} + 24 ⁣⋯ ⁣6724\!\cdots\!67T8+ T^{8} + 19 ⁣⋯ ⁣8419\!\cdots\!84T9+ T^{9} + 14 ⁣⋯ ⁣3214\!\cdots\!32T10+ T^{10} + 94 ⁣⋯ ⁣0094\!\cdots\!00T11+ T^{11} + 58 ⁣⋯ ⁣6458\!\cdots\!64T12+ T^{12} + 94 ⁣⋯ ⁣0094\!\cdots\!00p3T13+ p^{3} T^{13} + 14 ⁣⋯ ⁣3214\!\cdots\!32p6T14+ p^{6} T^{14} + 19 ⁣⋯ ⁣8419\!\cdots\!84p9T15+ p^{9} T^{15} + 24 ⁣⋯ ⁣6724\!\cdots\!67p12T16+ p^{12} T^{16} + 28 ⁣⋯ ⁣9228\!\cdots\!92p15T17+30580771234603876304p18T18+28876815994589208p21T19+23909532632490p24T20+16698964968p27T21+9472848p30T22+3888p33T23+p36T24 p^{15} T^{17} + 30580771234603876304 p^{18} T^{18} + 28876815994589208 p^{21} T^{19} + 23909532632490 p^{24} T^{20} + 16698964968 p^{27} T^{21} + 9472848 p^{30} T^{22} + 3888 p^{33} T^{23} + p^{36} T^{24}
73 1+1164T+2863680T2+2986228324T3+4116691297176T4+3722553087241332T5+3877761578271121090T6+ 1 + 1164 T + 2863680 T^{2} + 2986228324 T^{3} + 4116691297176 T^{4} + 3722553087241332 T^{5} + 3877761578271121090 T^{6} + 30 ⁣⋯ ⁣9630\!\cdots\!96T7+ T^{7} + 26 ⁣⋯ ⁣1226\!\cdots\!12T8+ T^{8} + 19 ⁣⋯ ⁣4419\!\cdots\!44T9+ T^{9} + 14 ⁣⋯ ⁣6414\!\cdots\!64T10+ T^{10} + 92 ⁣⋯ ⁣6492\!\cdots\!64T11+ T^{11} + 62 ⁣⋯ ⁣0762\!\cdots\!07T12+ T^{12} + 92 ⁣⋯ ⁣6492\!\cdots\!64p3T13+ p^{3} T^{13} + 14 ⁣⋯ ⁣6414\!\cdots\!64p6T14+ p^{6} T^{14} + 19 ⁣⋯ ⁣4419\!\cdots\!44p9T15+ p^{9} T^{15} + 26 ⁣⋯ ⁣1226\!\cdots\!12p12T16+ p^{12} T^{16} + 30 ⁣⋯ ⁣9630\!\cdots\!96p15T17+3877761578271121090p18T18+3722553087241332p21T19+4116691297176p24T20+2986228324p27T21+2863680p30T22+1164p33T23+p36T24 p^{15} T^{17} + 3877761578271121090 p^{18} T^{18} + 3722553087241332 p^{21} T^{19} + 4116691297176 p^{24} T^{20} + 2986228324 p^{27} T^{21} + 2863680 p^{30} T^{22} + 1164 p^{33} T^{23} + p^{36} T^{24}
79 1+2382T+6094200T2+9584500318T3+14733487983153T4+17532778836000066T5+20236880655873465859T6+ 1 + 2382 T + 6094200 T^{2} + 9584500318 T^{3} + 14733487983153 T^{4} + 17532778836000066 T^{5} + 20236880655873465859 T^{6} + 19 ⁣⋯ ⁣9619\!\cdots\!96T7+ T^{7} + 18 ⁣⋯ ⁣0118\!\cdots\!01T8+ T^{8} + 15 ⁣⋯ ⁣1015\!\cdots\!10T9+ T^{9} + 12 ⁣⋯ ⁣6912\!\cdots\!69T10+ T^{10} + 91 ⁣⋯ ⁣6091\!\cdots\!60T11+ T^{11} + 68 ⁣⋯ ⁣1868\!\cdots\!18T12+ T^{12} + 91 ⁣⋯ ⁣6091\!\cdots\!60p3T13+ p^{3} T^{13} + 12 ⁣⋯ ⁣6912\!\cdots\!69p6T14+ p^{6} T^{14} + 15 ⁣⋯ ⁣1015\!\cdots\!10p9T15+ p^{9} T^{15} + 18 ⁣⋯ ⁣0118\!\cdots\!01p12T16+ p^{12} T^{16} + 19 ⁣⋯ ⁣9619\!\cdots\!96p15T17+20236880655873465859p18T18+17532778836000066p21T19+14733487983153p24T20+9584500318p27T21+6094200p30T22+2382p33T23+p36T24 p^{15} T^{17} + 20236880655873465859 p^{18} T^{18} + 17532778836000066 p^{21} T^{19} + 14733487983153 p^{24} T^{20} + 9584500318 p^{27} T^{21} + 6094200 p^{30} T^{22} + 2382 p^{33} T^{23} + p^{36} T^{24}
83 1−4008T+10837092T2−21539693052T3+35491475230593T4−49939724231864238T5+62383132568798653751T6− 1 - 4008 T + 10837092 T^{2} - 21539693052 T^{3} + 35491475230593 T^{4} - 49939724231864238 T^{5} + 62383132568798653751 T^{6} - 70 ⁣⋯ ⁣4670\!\cdots\!46T7+ T^{7} + 72 ⁣⋯ ⁣7372\!\cdots\!73T8− T^{8} - 68 ⁣⋯ ⁣6668\!\cdots\!66T9+ T^{9} + 61 ⁣⋯ ⁣0961\!\cdots\!09T10− T^{10} - 50 ⁣⋯ ⁣5850\!\cdots\!58T11+ T^{11} + 39 ⁣⋯ ⁣9439\!\cdots\!94T12− T^{12} - 50 ⁣⋯ ⁣5850\!\cdots\!58p3T13+ p^{3} T^{13} + 61 ⁣⋯ ⁣0961\!\cdots\!09p6T14− p^{6} T^{14} - 68 ⁣⋯ ⁣6668\!\cdots\!66p9T15+ p^{9} T^{15} + 72 ⁣⋯ ⁣7372\!\cdots\!73p12T16− p^{12} T^{16} - 70 ⁣⋯ ⁣4670\!\cdots\!46p15T17+62383132568798653751p18T18−49939724231864238p21T19+35491475230593p24T20−21539693052p27T21+10837092p30T22−4008p33T23+p36T24 p^{15} T^{17} + 62383132568798653751 p^{18} T^{18} - 49939724231864238 p^{21} T^{19} + 35491475230593 p^{24} T^{20} - 21539693052 p^{27} T^{21} + 10837092 p^{30} T^{22} - 4008 p^{33} T^{23} + p^{36} T^{24}
89 1−3582T+9208023T2−16371231642T3+24664465339941T4−30799525034820456T5+35164856630730158042T6− 1 - 3582 T + 9208023 T^{2} - 16371231642 T^{3} + 24664465339941 T^{4} - 30799525034820456 T^{5} + 35164856630730158042 T^{6} - 35 ⁣⋯ ⁣6635\!\cdots\!66T7+ T^{7} + 35 ⁣⋯ ⁣5435\!\cdots\!54T8− T^{8} - 32 ⁣⋯ ⁣2832\!\cdots\!28T9+ T^{9} + 28 ⁣⋯ ⁣7828\!\cdots\!78T10− T^{10} - 24 ⁣⋯ ⁣8424\!\cdots\!84T11+ T^{11} + 20 ⁣⋯ ⁣3120\!\cdots\!31T12− T^{12} - 24 ⁣⋯ ⁣8424\!\cdots\!84p3T13+ p^{3} T^{13} + 28 ⁣⋯ ⁣7828\!\cdots\!78p6T14− p^{6} T^{14} - 32 ⁣⋯ ⁣2832\!\cdots\!28p9T15+ p^{9} T^{15} + 35 ⁣⋯ ⁣5435\!\cdots\!54p12T16− p^{12} T^{16} - 35 ⁣⋯ ⁣6635\!\cdots\!66p15T17+35164856630730158042p18T18−30799525034820456p21T19+24664465339941p24T20−16371231642p27T21+9208023p30T22−3582p33T23+p36T24 p^{15} T^{17} + 35164856630730158042 p^{18} T^{18} - 30799525034820456 p^{21} T^{19} + 24664465339941 p^{24} T^{20} - 16371231642 p^{27} T^{21} + 9208023 p^{30} T^{22} - 3582 p^{33} T^{23} + p^{36} T^{24}
97 1+2958T+9267033T2+17310905872T3+32258081601246T4+45739676328280764T5+65493331632543782548T6+ 1 + 2958 T + 9267033 T^{2} + 17310905872 T^{3} + 32258081601246 T^{4} + 45739676328280764 T^{5} + 65493331632543782548 T^{6} + 78 ⁣⋯ ⁣6478\!\cdots\!64T7+ T^{7} + 97 ⁣⋯ ⁣9797\!\cdots\!97T8+ T^{8} + 10 ⁣⋯ ⁣4010\!\cdots\!40T9+ T^{9} + 11 ⁣⋯ ⁣7511\!\cdots\!75T10+ T^{10} + 11 ⁣⋯ ⁣3011\!\cdots\!30T11+ T^{11} + 11 ⁣⋯ ⁣9311\!\cdots\!93T12+ T^{12} + 11 ⁣⋯ ⁣3011\!\cdots\!30p3T13+ p^{3} T^{13} + 11 ⁣⋯ ⁣7511\!\cdots\!75p6T14+ p^{6} T^{14} + 10 ⁣⋯ ⁣4010\!\cdots\!40p9T15+ p^{9} T^{15} + 97 ⁣⋯ ⁣9797\!\cdots\!97p12T16+ p^{12} T^{16} + 78 ⁣⋯ ⁣6478\!\cdots\!64p15T17+65493331632543782548p18T18+45739676328280764p21T19+32258081601246p24T20+17310905872p27T21+9267033p30T22+2958p33T23+p36T24 p^{15} T^{17} + 65493331632543782548 p^{18} T^{18} + 45739676328280764 p^{21} T^{19} + 32258081601246 p^{24} T^{20} + 17310905872 p^{27} T^{21} + 9267033 p^{30} T^{22} + 2958 p^{33} T^{23} + p^{36} T^{24}
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   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.42929911007811144862292066857, −3.38885142124257634644467567073, −3.37592572394990817215756295684, −3.21590534241337533782720449191, −3.08705953444681537812423322391, −2.92023081437122245038354063608, −2.91174997146627189071504703332, −2.86118011775947796808609270899, −2.67016683419126260393896804244, −2.54987245452327908161607043991, −2.52866965430446396264905710432, −2.38450245561170926897565625498, −2.28313245796351115196952545098, −2.25946163958332616887493508798, −2.02881911936424057624363202624, −1.89867807127348884656450433032, −1.54752154362656577012885012450, −1.53218522790649262392809046962, −1.52161934914990450043539727985, −1.52141209927975624102735209899, −1.29995900300356915846892043289, −1.16582936168310692570938064062, −1.11033080270694175526676275209, −1.08146117700454714254201836128, −0.867936414641235139284565861962, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.867936414641235139284565861962, 1.08146117700454714254201836128, 1.11033080270694175526676275209, 1.16582936168310692570938064062, 1.29995900300356915846892043289, 1.52141209927975624102735209899, 1.52161934914990450043539727985, 1.53218522790649262392809046962, 1.54752154362656577012885012450, 1.89867807127348884656450433032, 2.02881911936424057624363202624, 2.25946163958332616887493508798, 2.28313245796351115196952545098, 2.38450245561170926897565625498, 2.52866965430446396264905710432, 2.54987245452327908161607043991, 2.67016683419126260393896804244, 2.86118011775947796808609270899, 2.91174997146627189071504703332, 2.92023081437122245038354063608, 3.08705953444681537812423322391, 3.21590534241337533782720449191, 3.37592572394990817215756295684, 3.38885142124257634644467567073, 3.42929911007811144862292066857

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.