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Properties

Label 40-160e20-1.1-c2e20-0-0
Degree 4040
Conductor 1.209×10441.209\times 10^{44}
Sign 11
Analytic cond. 6.15319×10126.15319\times 10^{12}
Root an. cond. 2.087982.08798
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 12·17-s + 4·23-s − 14·25-s + 136·31-s − 8·41-s − 188·47-s + 8·49-s − 248·71-s − 124·73-s + 114·81-s + 100·97-s + 516·103-s − 284·113-s − 48·119-s + 1.31e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + 167-s + 173-s + ⋯
L(s)  = 1  + 4/7·7-s − 0.705·17-s + 4/23·23-s − 0.559·25-s + 4.38·31-s − 0.195·41-s − 4·47-s + 8/49·49-s − 3.49·71-s − 1.69·73-s + 1.40·81-s + 1.03·97-s + 5.00·103-s − 2.51·113-s − 0.403·119-s + 10.8·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.0993·161-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + ⋯

Functional equation

Λ(s)=((2100⋅520)s/2 ΓC(s)20 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((2100⋅520)s/2 ΓC(s+1)20 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{100} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 4040
Conductor: 2100⋅5202^{100} \cdot 5^{20}
Sign: 11
Analytic conductor: 6.15319×10126.15319\times 10^{12}
Root analytic conductor: 2.087982.08798
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (40, 2100⋅520, ( :[1]20), 1)(40,\ 2^{100} \cdot 5^{20} ,\ ( \ : [1]^{20} ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.1882867573.188286757
L(12)L(\frac12) ≈\approx 3.1882867573.188286757
L(2)L(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)
bad2 1 1
5 1+14T2+41pT4−2808T6+2034p2T8+8084p4T10+2034p6T12−2808p8T14+41p13T16+14p16T18+p20T20 1 + 14 T^{2} + 41 p T^{4} - 2808 T^{6} + 2034 p^{2} T^{8} + 8084 p^{4} T^{10} + 2034 p^{6} T^{12} - 2808 p^{8} T^{14} + 41 p^{13} T^{16} + 14 p^{16} T^{18} + p^{20} T^{20}
good3 1−38pT4−1859T8+1114504T12−14869022T16−50382700p4T20−14869022p8T24+1114504p16T28−1859p24T32−38p33T36+p40T40 1 - 38 p T^{4} - 1859 T^{8} + 1114504 T^{12} - 14869022 T^{16} - 50382700 p^{4} T^{20} - 14869022 p^{8} T^{24} + 1114504 p^{16} T^{28} - 1859 p^{24} T^{32} - 38 p^{33} T^{36} + p^{40} T^{40}
7 (1−2T+2T2−610T3+271pT4+18320T5+145616T6−1137328T7−8045434T8+18540420T9+564357756T10+18540420p2T11−8045434p4T12−1137328p6T13+145616p8T14+18320p10T15+271p13T16−610p14T17+2p16T18−2p18T19+p20T20)2 ( 1 - 2 T + 2 T^{2} - 610 T^{3} + 271 p T^{4} + 18320 T^{5} + 145616 T^{6} - 1137328 T^{7} - 8045434 T^{8} + 18540420 T^{9} + 564357756 T^{10} + 18540420 p^{2} T^{11} - 8045434 p^{4} T^{12} - 1137328 p^{6} T^{13} + 145616 p^{8} T^{14} + 18320 p^{10} T^{15} + 271 p^{13} T^{16} - 610 p^{14} T^{17} + 2 p^{16} T^{18} - 2 p^{18} T^{19} + p^{20} T^{20} )^{2}
11 (1−658T2+224781T4−52340600T6+9120139298T8−1240167831404T10+9120139298p4T12−52340600p8T14+224781p12T16−658p16T18+p20T20)2 ( 1 - 658 T^{2} + 224781 T^{4} - 52340600 T^{6} + 9120139298 T^{8} - 1240167831404 T^{10} + 9120139298 p^{4} T^{12} - 52340600 p^{8} T^{14} + 224781 p^{12} T^{16} - 658 p^{16} T^{18} + p^{20} T^{20} )^{2}
13 1−68714T4+3087675853T8−85473147028472T12+2398905635263377746T16− 1 - 68714 T^{4} + 3087675853 T^{8} - 85473147028472 T^{12} + 2398905635263377746 T^{16} - 60 ⁣⋯ ⁣2860\!\cdots\!28T20+2398905635263377746p8T24−85473147028472p16T28+3087675853p24T32−68714p32T36+p40T40 T^{20} + 2398905635263377746 p^{8} T^{24} - 85473147028472 p^{16} T^{28} + 3087675853 p^{24} T^{32} - 68714 p^{32} T^{36} + p^{40} T^{40}
17 (1+6T+18T2+12134T3+9389pT4−1173048T5+63705656T6+1574926728T7−7035754814T8+54207870724T9+10081821662572T10+54207870724p2T11−7035754814p4T12+1574926728p6T13+63705656p8T14−1173048p10T15+9389p13T16+12134p14T17+18p16T18+6p18T19+p20T20)2 ( 1 + 6 T + 18 T^{2} + 12134 T^{3} + 9389 p T^{4} - 1173048 T^{5} + 63705656 T^{6} + 1574926728 T^{7} - 7035754814 T^{8} + 54207870724 T^{9} + 10081821662572 T^{10} + 54207870724 p^{2} T^{11} - 7035754814 p^{4} T^{12} + 1574926728 p^{6} T^{13} + 63705656 p^{8} T^{14} - 1173048 p^{10} T^{15} + 9389 p^{13} T^{16} + 12134 p^{14} T^{17} + 18 p^{16} T^{18} + 6 p^{18} T^{19} + p^{20} T^{20} )^{2}
19 (1+2406T2+2930557T4+2315128904T6+1303230895506T8+543406660187172T10+1303230895506p4T12+2315128904p8T14+2930557p12T16+2406p16T18+p20T20)2 ( 1 + 2406 T^{2} + 2930557 T^{4} + 2315128904 T^{6} + 1303230895506 T^{8} + 543406660187172 T^{10} + 1303230895506 p^{4} T^{12} + 2315128904 p^{8} T^{14} + 2930557 p^{12} T^{16} + 2406 p^{16} T^{18} + p^{20} T^{20} )^{2}
23 (1−2T+2T2+2494T3+280233T4−2455824T5+324400pT6+589949040T7−128728992634T8−513870857180T9+3681193905276T10−513870857180p2T11−128728992634p4T12+589949040p6T13+324400p9T14−2455824p10T15+280233p12T16+2494p14T17+2p16T18−2p18T19+p20T20)2 ( 1 - 2 T + 2 T^{2} + 2494 T^{3} + 280233 T^{4} - 2455824 T^{5} + 324400 p T^{6} + 589949040 T^{7} - 128728992634 T^{8} - 513870857180 T^{9} + 3681193905276 T^{10} - 513870857180 p^{2} T^{11} - 128728992634 p^{4} T^{12} + 589949040 p^{6} T^{13} + 324400 p^{9} T^{14} - 2455824 p^{10} T^{15} + 280233 p^{12} T^{16} + 2494 p^{14} T^{17} + 2 p^{16} T^{18} - 2 p^{18} T^{19} + p^{20} T^{20} )^{2}
29 (1+5862T2+16207421T4+28469762120T6+35887152044178T8+34347575552131876T10+35887152044178p4T12+28469762120p8T14+16207421p12T16+5862p16T18+p20T20)2 ( 1 + 5862 T^{2} + 16207421 T^{4} + 28469762120 T^{6} + 35887152044178 T^{8} + 34347575552131876 T^{10} + 35887152044178 p^{4} T^{12} + 28469762120 p^{8} T^{14} + 16207421 p^{12} T^{16} + 5862 p^{16} T^{18} + p^{20} T^{20} )^{2}
31 (1−34T+119pT2−73552T3+5377158T4−77314812T5+5377158p2T6−73552p4T7+119p7T8−34p8T9+p10T10)4 ( 1 - 34 T + 119 p T^{2} - 73552 T^{3} + 5377158 T^{4} - 77314812 T^{5} + 5377158 p^{2} T^{6} - 73552 p^{4} T^{7} + 119 p^{7} T^{8} - 34 p^{8} T^{9} + p^{10} T^{10} )^{4}
37 1+4173782T4+12570586938765T8+36325113880823571720T12+ 1 + 4173782 T^{4} + 12570586938765 T^{8} + 36325113880823571720 T^{12} + 81 ⁣⋯ ⁣3881\!\cdots\!38T16+ T^{16} + 15 ⁣⋯ ⁣6815\!\cdots\!68T20+ T^{20} + 81 ⁣⋯ ⁣3881\!\cdots\!38p8T24+36325113880823571720p16T28+12570586938765p24T32+4173782p32T36+p40T40 p^{8} T^{24} + 36325113880823571720 p^{16} T^{28} + 12570586938765 p^{24} T^{32} + 4173782 p^{32} T^{36} + p^{40} T^{40}
41 (1+2T+3929T2−43776T3+9954182T4−83797156T5+9954182p2T6−43776p4T7+3929p6T8+2p8T9+p10T10)4 ( 1 + 2 T + 3929 T^{2} - 43776 T^{3} + 9954182 T^{4} - 83797156 T^{5} + 9954182 p^{2} T^{6} - 43776 p^{4} T^{7} + 3929 p^{6} T^{8} + 2 p^{8} T^{9} + p^{10} T^{10} )^{4}
43 1−3802546T4+26831738947197T8−91894552069346003832T12+ 1 - 3802546 T^{4} + 26831738947197 T^{8} - 91894552069346003832 T^{12} + 50 ⁣⋯ ⁣7050\!\cdots\!70T16− T^{16} - 13 ⁣⋯ ⁣8013\!\cdots\!80T20+ T^{20} + 50 ⁣⋯ ⁣7050\!\cdots\!70p8T24−91894552069346003832p16T28+26831738947197p24T32−3802546p32T36+p40T40 p^{8} T^{24} - 91894552069346003832 p^{16} T^{28} + 26831738947197 p^{24} T^{32} - 3802546 p^{32} T^{36} + p^{40} T^{40}
47 (1+2pT+2p2T2+248926T3+12120201T4+392566800T5+14336307920T6+634261663440T7+14862450450758T8+118302473355652T9+6637757880275964T10+118302473355652p2T11+14862450450758p4T12+634261663440p6T13+14336307920p8T14+392566800p10T15+12120201p12T16+248926p14T17+2p18T18+2p19T19+p20T20)2 ( 1 + 2 p T + 2 p^{2} T^{2} + 248926 T^{3} + 12120201 T^{4} + 392566800 T^{5} + 14336307920 T^{6} + 634261663440 T^{7} + 14862450450758 T^{8} + 118302473355652 T^{9} + 6637757880275964 T^{10} + 118302473355652 p^{2} T^{11} + 14862450450758 p^{4} T^{12} + 634261663440 p^{6} T^{13} + 14336307920 p^{8} T^{14} + 392566800 p^{10} T^{15} + 12120201 p^{12} T^{16} + 248926 p^{14} T^{17} + 2 p^{18} T^{18} + 2 p^{19} T^{19} + p^{20} T^{20} )^{2}
53 1−15039146T4+133684220395021T8+39218946710640122632T12− 1 - 15039146 T^{4} + 133684220395021 T^{8} + 39218946710640122632 T^{12} - 96 ⁣⋯ ⁣0296\!\cdots\!02T16+ T^{16} + 12 ⁣⋯ ⁣2812\!\cdots\!28T20− T^{20} - 96 ⁣⋯ ⁣0296\!\cdots\!02p8T24+39218946710640122632p16T28+133684220395021p24T32−15039146p32T36+p40T40 p^{8} T^{24} + 39218946710640122632 p^{16} T^{28} + 133684220395021 p^{24} T^{32} - 15039146 p^{32} T^{36} + p^{40} T^{40}
59 (1+20358T2+205085277T4+1346166325448T6+6536734169700626T8+25203640704687976420T10+6536734169700626p4T12+1346166325448p8T14+205085277p12T16+20358p16T18+p20T20)2 ( 1 + 20358 T^{2} + 205085277 T^{4} + 1346166325448 T^{6} + 6536734169700626 T^{8} + 25203640704687976420 T^{10} + 6536734169700626 p^{4} T^{12} + 1346166325448 p^{8} T^{14} + 205085277 p^{12} T^{16} + 20358 p^{16} T^{18} + p^{20} T^{20} )^{2}
61 (1−18802T2+189117357T4−1333577301752T6+7106541440264866T8−29659798453891124140T10+7106541440264866p4T12−1333577301752p8T14+189117357p12T16−18802p16T18+p20T20)2 ( 1 - 18802 T^{2} + 189117357 T^{4} - 1333577301752 T^{6} + 7106541440264866 T^{8} - 29659798453891124140 T^{10} + 7106541440264866 p^{4} T^{12} - 1333577301752 p^{8} T^{14} + 189117357 p^{12} T^{16} - 18802 p^{16} T^{18} + p^{20} T^{20} )^{2}
67 1+6517774T4+162116992238781T8− 1 + 6517774 T^{4} + 162116992238781 T^{8} - 33 ⁣⋯ ⁣0833\!\cdots\!08T12+ T^{12} + 23 ⁣⋯ ⁣9823\!\cdots\!98T16+ T^{16} + 28 ⁣⋯ ⁣6828\!\cdots\!68T20+ T^{20} + 23 ⁣⋯ ⁣9823\!\cdots\!98p8T24− p^{8} T^{24} - 33 ⁣⋯ ⁣0833\!\cdots\!08p16T28+162116992238781p24T32+6517774p32T36+p40T40 p^{16} T^{28} + 162116992238781 p^{24} T^{32} + 6517774 p^{32} T^{36} + p^{40} T^{40}
71 (1+62T+12585T2+1087696T3+91333814T4+7391992324T5+91333814p2T6+1087696p4T7+12585p6T8+62p8T9+p10T10)4 ( 1 + 62 T + 12585 T^{2} + 1087696 T^{3} + 91333814 T^{4} + 7391992324 T^{5} + 91333814 p^{2} T^{6} + 1087696 p^{4} T^{7} + 12585 p^{6} T^{8} + 62 p^{8} T^{9} + p^{10} T^{10} )^{4}
73 (1+62T+1922T2+293406T3+52885149T4+1992907656T5+64958558712T6+9681205197576T7+2398655765914626T8+112307032540117748T9+3743840733961712140T10+112307032540117748p2T11+2398655765914626p4T12+9681205197576p6T13+64958558712p8T14+1992907656p10T15+52885149p12T16+293406p14T17+1922p16T18+62p18T19+p20T20)2 ( 1 + 62 T + 1922 T^{2} + 293406 T^{3} + 52885149 T^{4} + 1992907656 T^{5} + 64958558712 T^{6} + 9681205197576 T^{7} + 2398655765914626 T^{8} + 112307032540117748 T^{9} + 3743840733961712140 T^{10} + 112307032540117748 p^{2} T^{11} + 2398655765914626 p^{4} T^{12} + 9681205197576 p^{6} T^{13} + 64958558712 p^{8} T^{14} + 1992907656 p^{10} T^{15} + 52885149 p^{12} T^{16} + 293406 p^{14} T^{17} + 1922 p^{16} T^{18} + 62 p^{18} T^{19} + p^{20} T^{20} )^{2}
79 (1−46346T2+1023688173T4−14286438751864T6+140168501571945426T8− ( 1 - 46346 T^{2} + 1023688173 T^{4} - 14286438751864 T^{6} + 140168501571945426 T^{8} - 10 ⁣⋯ ⁣2010\!\cdots\!20T10+140168501571945426p4T12−14286438751864p8T14+1023688173p12T16−46346p16T18+p20T20)2 T^{10} + 140168501571945426 p^{4} T^{12} - 14286438751864 p^{8} T^{14} + 1023688173 p^{12} T^{16} - 46346 p^{16} T^{18} + p^{20} T^{20} )^{2}
83 1−209971186T4+203835292037679pT8− 1 - 209971186 T^{4} + 203835292037679 p T^{8} - 34 ⁣⋯ ⁣5234\!\cdots\!52T12− T^{12} - 43 ⁣⋯ ⁣9043\!\cdots\!90T16+ T^{16} + 36 ⁣⋯ ⁣2036\!\cdots\!20T20− T^{20} - 43 ⁣⋯ ⁣9043\!\cdots\!90p8T24− p^{8} T^{24} - 34 ⁣⋯ ⁣5234\!\cdots\!52p16T28+203835292037679p25T32−209971186p32T36+p40T40 p^{16} T^{28} + 203835292037679 p^{25} T^{32} - 209971186 p^{32} T^{36} + p^{40} T^{40}
89 (1−46986T2+1163729293T4−215914071096pT6+230705121354497106T8− ( 1 - 46986 T^{2} + 1163729293 T^{4} - 215914071096 p T^{6} + 230705121354497106 T^{8} - 20 ⁣⋯ ⁣6020\!\cdots\!60T10+230705121354497106p4T12−215914071096p9T14+1163729293p12T16−46986p16T18+p20T20)2 T^{10} + 230705121354497106 p^{4} T^{12} - 215914071096 p^{9} T^{14} + 1163729293 p^{12} T^{16} - 46986 p^{16} T^{18} + p^{20} T^{20} )^{2}
97 (1−50T+1250T2−914226T3+90832893T4+7094951240T5−50384088712T6+11147294809672T7−3740546910279294T8+653351137546335316T9−8437320896266469556T10+653351137546335316p2T11−3740546910279294p4T12+11147294809672p6T13−50384088712p8T14+7094951240p10T15+90832893p12T16−914226p14T17+1250p16T18−50p18T19+p20T20)2 ( 1 - 50 T + 1250 T^{2} - 914226 T^{3} + 90832893 T^{4} + 7094951240 T^{5} - 50384088712 T^{6} + 11147294809672 T^{7} - 3740546910279294 T^{8} + 653351137546335316 T^{9} - 8437320896266469556 T^{10} + 653351137546335316 p^{2} T^{11} - 3740546910279294 p^{4} T^{12} + 11147294809672 p^{6} T^{13} - 50384088712 p^{8} T^{14} + 7094951240 p^{10} T^{15} + 90832893 p^{12} T^{16} - 914226 p^{14} T^{17} + 1250 p^{16} T^{18} - 50 p^{18} T^{19} + p^{20} T^{20} )^{2}
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   L(s)=∏p ∏j=140(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.02878896273069215571353086641, −2.92930207216142546411213077661, −2.91538082145007012691973105096, −2.69039405599357588105941227665, −2.65370403351739257986545832013, −2.63129950264265721849820806214, −2.42302226682149180536455567434, −2.40411754626911255726855711376, −2.36247294702625153469859971215, −2.24651062898452295193772436094, −2.00297783892652583735740341010, −1.95496074922009698580550575625, −1.78897653077391960579341822836, −1.74278947917878184215171573022, −1.71192332277655219207281132029, −1.61223040739077806774761123099, −1.46873799985940028876846820852, −1.39407383235160307719312885845, −1.05368837344791461721012531743, −0.903583637409107382885236592608, −0.872981040050345257121315903951, −0.76425047923520404724294204667, −0.58762804741124329717946617472, −0.28428959623764864941835423069, −0.12879466634240432482580080998, 0.12879466634240432482580080998, 0.28428959623764864941835423069, 0.58762804741124329717946617472, 0.76425047923520404724294204667, 0.872981040050345257121315903951, 0.903583637409107382885236592608, 1.05368837344791461721012531743, 1.39407383235160307719312885845, 1.46873799985940028876846820852, 1.61223040739077806774761123099, 1.71192332277655219207281132029, 1.74278947917878184215171573022, 1.78897653077391960579341822836, 1.95496074922009698580550575625, 2.00297783892652583735740341010, 2.24651062898452295193772436094, 2.36247294702625153469859971215, 2.40411754626911255726855711376, 2.42302226682149180536455567434, 2.63129950264265721849820806214, 2.65370403351739257986545832013, 2.69039405599357588105941227665, 2.91538082145007012691973105096, 2.92930207216142546411213077661, 3.02878896273069215571353086641

Graph of the ZZ-function along the critical line

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