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Properties

Label 20-325e10-1.1-c1e10-0-2
Degree 2020
Conductor 1.315×10251.315\times 10^{25}
Sign 11
Analytic cond. 13854.913854.9
Root an. cond. 1.610941.61094
Motivic weight 11
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  − 3·3-s + 2·4-s + 2·7-s + 2·8-s + 10·9-s − 3·11-s − 6·12-s + 10·13-s + 4·16-s − 4·17-s + 4·19-s − 6·21-s − 15·23-s − 6·24-s − 13·27-s + 4·28-s + 29-s − 32-s + 9·33-s + 20·36-s − 17·37-s − 30·39-s − 6·41-s − 12·43-s − 6·44-s − 24·47-s − 12·48-s + ⋯
L(s)  = 1  − 1.73·3-s + 4-s + 0.755·7-s + 0.707·8-s + 10/3·9-s − 0.904·11-s − 1.73·12-s + 2.77·13-s + 16-s − 0.970·17-s + 0.917·19-s − 1.30·21-s − 3.12·23-s − 1.22·24-s − 2.50·27-s + 0.755·28-s + 0.185·29-s − 0.176·32-s + 1.56·33-s + 10/3·36-s − 2.79·37-s − 4.80·39-s − 0.937·41-s − 1.82·43-s − 0.904·44-s − 3.50·47-s − 1.73·48-s + ⋯

Functional equation

Λ(s)=((520⋅1310)s/2 ΓC(s)10 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((520⋅1310)s/2 ΓC(s+1/2)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 520⋅13105^{20} \cdot 13^{10}
Sign: 11
Analytic conductor: 13854.913854.9
Root analytic conductor: 1.610941.61094
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 520⋅1310, ( :[1/2]10), 1)(20,\ 5^{20} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.5034698172.503469817
L(12)L(\frac12) ≈\approx 2.5034698172.503469817
L(32)L(\frac{3}{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)
bad5 1 1
13 1−10T+60T2−263T3+1076T4−3957T5+1076pT6−263p2T7+60p3T8−10p4T9+p5T10 1 - 10 T + 60 T^{2} - 263 T^{3} + 1076 T^{4} - 3957 T^{5} + 1076 p T^{6} - 263 p^{2} T^{7} + 60 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10}
good2 1−pT2−pT3+9T5+9T6−5p2T7−23T8+5p2T9+45T10+5p3T11−23p2T12−5p5T13+9p4T14+9p5T15−p8T17−p9T18+p10T20 1 - p T^{2} - p T^{3} + 9 T^{5} + 9 T^{6} - 5 p^{2} T^{7} - 23 T^{8} + 5 p^{2} T^{9} + 45 T^{10} + 5 p^{3} T^{11} - 23 p^{2} T^{12} - 5 p^{5} T^{13} + 9 p^{4} T^{14} + 9 p^{5} T^{15} - p^{8} T^{17} - p^{9} T^{18} + p^{10} T^{20}
3 1+pT−T2−20T3−26T4+41T5+130T6+19T7−274T8−43pT9+409T10−43p2T11−274p2T12+19p3T13+130p4T14+41p5T15−26p6T16−20p7T17−p8T18+p10T19+p10T20 1 + p T - T^{2} - 20 T^{3} - 26 T^{4} + 41 T^{5} + 130 T^{6} + 19 T^{7} - 274 T^{8} - 43 p T^{9} + 409 T^{10} - 43 p^{2} T^{11} - 274 p^{2} T^{12} + 19 p^{3} T^{13} + 130 p^{4} T^{14} + 41 p^{5} T^{15} - 26 p^{6} T^{16} - 20 p^{7} T^{17} - p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20}
7 1−2T−12T2−8T3+97T4+281T5−260T6−1954T7−835T8+2855T9+19063T10+2855pT11−835p2T12−1954p3T13−260p4T14+281p5T15+97p6T16−8p7T17−12p8T18−2p9T19+p10T20 1 - 2 T - 12 T^{2} - 8 T^{3} + 97 T^{4} + 281 T^{5} - 260 T^{6} - 1954 T^{7} - 835 T^{8} + 2855 T^{9} + 19063 T^{10} + 2855 p T^{11} - 835 p^{2} T^{12} - 1954 p^{3} T^{13} - 260 p^{4} T^{14} + 281 p^{5} T^{15} + 97 p^{6} T^{16} - 8 p^{7} T^{17} - 12 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20}
11 1+3T−29T2−38T3+576T4−3T5−7311T6+6832T7+74554T8−49384T9−752787T10−49384pT11+74554p2T12+6832p3T13−7311p4T14−3p5T15+576p6T16−38p7T17−29p8T18+3p9T19+p10T20 1 + 3 T - 29 T^{2} - 38 T^{3} + 576 T^{4} - 3 T^{5} - 7311 T^{6} + 6832 T^{7} + 74554 T^{8} - 49384 T^{9} - 752787 T^{10} - 49384 p T^{11} + 74554 p^{2} T^{12} + 6832 p^{3} T^{13} - 7311 p^{4} T^{14} - 3 p^{5} T^{15} + 576 p^{6} T^{16} - 38 p^{7} T^{17} - 29 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20}
17 1+4T−33T2−158T3+411T4+2612T5−4681T6−45867T7+1590T8+464191T9+1440069T10+464191pT11+1590p2T12−45867p3T13−4681p4T14+2612p5T15+411p6T16−158p7T17−33p8T18+4p9T19+p10T20 1 + 4 T - 33 T^{2} - 158 T^{3} + 411 T^{4} + 2612 T^{5} - 4681 T^{6} - 45867 T^{7} + 1590 T^{8} + 464191 T^{9} + 1440069 T^{10} + 464191 p T^{11} + 1590 p^{2} T^{12} - 45867 p^{3} T^{13} - 4681 p^{4} T^{14} + 2612 p^{5} T^{15} + 411 p^{6} T^{16} - 158 p^{7} T^{17} - 33 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20}
19 1−4T−71T2+194T3+3329T4−5754T5−5837pT6+98987T7+2917208T8−788473T9−61357653T10−788473pT11+2917208p2T12+98987p3T13−5837p5T14−5754p5T15+3329p6T16+194p7T17−71p8T18−4p9T19+p10T20 1 - 4 T - 71 T^{2} + 194 T^{3} + 3329 T^{4} - 5754 T^{5} - 5837 p T^{6} + 98987 T^{7} + 2917208 T^{8} - 788473 T^{9} - 61357653 T^{10} - 788473 p T^{11} + 2917208 p^{2} T^{12} + 98987 p^{3} T^{13} - 5837 p^{5} T^{14} - 5754 p^{5} T^{15} + 3329 p^{6} T^{16} + 194 p^{7} T^{17} - 71 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20}
23 1+15T+82T2+51T3−1455T4−4488T5+18491T6+99225T7−338668T8−4764405T9−26762865T10−4764405pT11−338668p2T12+99225p3T13+18491p4T14−4488p5T15−1455p6T16+51p7T17+82p8T18+15p9T19+p10T20 1 + 15 T + 82 T^{2} + 51 T^{3} - 1455 T^{4} - 4488 T^{5} + 18491 T^{6} + 99225 T^{7} - 338668 T^{8} - 4764405 T^{9} - 26762865 T^{10} - 4764405 p T^{11} - 338668 p^{2} T^{12} + 99225 p^{3} T^{13} + 18491 p^{4} T^{14} - 4488 p^{5} T^{15} - 1455 p^{6} T^{16} + 51 p^{7} T^{17} + 82 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20}
29 1−T−79T2−146T3+3192T4+11779T5−67556T6−384403T7+890654T8+4091745T9−7513779T10+4091745pT11+890654p2T12−384403p3T13−67556p4T14+11779p5T15+3192p6T16−146p7T17−79p8T18−p9T19+p10T20 1 - T - 79 T^{2} - 146 T^{3} + 3192 T^{4} + 11779 T^{5} - 67556 T^{6} - 384403 T^{7} + 890654 T^{8} + 4091745 T^{9} - 7513779 T^{10} + 4091745 p T^{11} + 890654 p^{2} T^{12} - 384403 p^{3} T^{13} - 67556 p^{4} T^{14} + 11779 p^{5} T^{15} + 3192 p^{6} T^{16} - 146 p^{7} T^{17} - 79 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20}
31 (1+98T2−29T3+4804T4−1573T5+4804pT6−29p2T7+98p3T8+p5T10)2 ( 1 + 98 T^{2} - 29 T^{3} + 4804 T^{4} - 1573 T^{5} + 4804 p T^{6} - 29 p^{2} T^{7} + 98 p^{3} T^{8} + p^{5} T^{10} )^{2}
37 1+17T+51T2−502T3−372T4+37861T5+106407T6−1104416T7−5601148T8+10960902T9+176313507T10+10960902pT11−5601148p2T12−1104416p3T13+106407p4T14+37861p5T15−372p6T16−502p7T17+51p8T18+17p9T19+p10T20 1 + 17 T + 51 T^{2} - 502 T^{3} - 372 T^{4} + 37861 T^{5} + 106407 T^{6} - 1104416 T^{7} - 5601148 T^{8} + 10960902 T^{9} + 176313507 T^{10} + 10960902 p T^{11} - 5601148 p^{2} T^{12} - 1104416 p^{3} T^{13} + 106407 p^{4} T^{14} + 37861 p^{5} T^{15} - 372 p^{6} T^{16} - 502 p^{7} T^{17} + 51 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20}
41 1+6T−95T2−384T3+6327T4+16770T5−184789T6−272397T7+682286T8+2994381T9+158651253T10+2994381pT11+682286p2T12−272397p3T13−184789p4T14+16770p5T15+6327p6T16−384p7T17−95p8T18+6p9T19+p10T20 1 + 6 T - 95 T^{2} - 384 T^{3} + 6327 T^{4} + 16770 T^{5} - 184789 T^{6} - 272397 T^{7} + 682286 T^{8} + 2994381 T^{9} + 158651253 T^{10} + 2994381 p T^{11} + 682286 p^{2} T^{12} - 272397 p^{3} T^{13} - 184789 p^{4} T^{14} + 16770 p^{5} T^{15} + 6327 p^{6} T^{16} - 384 p^{7} T^{17} - 95 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20}
43 1+12T−17T2−1046T3−4829T4+18006T5+201631T6+9867pT7−166530T8−8309917T9−106017329T10−8309917pT11−166530p2T12+9867p4T13+201631p4T14+18006p5T15−4829p6T16−1046p7T17−17p8T18+12p9T19+p10T20 1 + 12 T - 17 T^{2} - 1046 T^{3} - 4829 T^{4} + 18006 T^{5} + 201631 T^{6} + 9867 p T^{7} - 166530 T^{8} - 8309917 T^{9} - 106017329 T^{10} - 8309917 p T^{11} - 166530 p^{2} T^{12} + 9867 p^{4} T^{13} + 201631 p^{4} T^{14} + 18006 p^{5} T^{15} - 4829 p^{6} T^{16} - 1046 p^{7} T^{17} - 17 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20}
47 (1+12T+215T2+1945T3+19360T4+131839T5+19360pT6+1945p2T7+215p3T8+12p4T9+p5T10)2 ( 1 + 12 T + 215 T^{2} + 1945 T^{3} + 19360 T^{4} + 131839 T^{5} + 19360 p T^{6} + 1945 p^{2} T^{7} + 215 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2}
53 (1−8T+148T2−1417T3+13132T4−99183T5+13132pT6−1417p2T7+148p3T8−8p4T9+p5T10)2 ( 1 - 8 T + 148 T^{2} - 1417 T^{3} + 13132 T^{4} - 99183 T^{5} + 13132 p T^{6} - 1417 p^{2} T^{7} + 148 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2}
59 1−12T−64T2+738T3+96pT4−9081T5−336339T6−2403693T7+23185887T8+87777036T9−1346009523T10+87777036pT11+23185887p2T12−2403693p3T13−336339p4T14−9081p5T15+96p7T16+738p7T17−64p8T18−12p9T19+p10T20 1 - 12 T - 64 T^{2} + 738 T^{3} + 96 p T^{4} - 9081 T^{5} - 336339 T^{6} - 2403693 T^{7} + 23185887 T^{8} + 87777036 T^{9} - 1346009523 T^{10} + 87777036 p T^{11} + 23185887 p^{2} T^{12} - 2403693 p^{3} T^{13} - 336339 p^{4} T^{14} - 9081 p^{5} T^{15} + 96 p^{7} T^{16} + 738 p^{7} T^{17} - 64 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20}
61 1+5T−233T2−770T3+32534T4+66025T5−3312205T6−3580430T7+267548612T8+86879620T9−17862005833T10+86879620pT11+267548612p2T12−3580430p3T13−3312205p4T14+66025p5T15+32534p6T16−770p7T17−233p8T18+5p9T19+p10T20 1 + 5 T - 233 T^{2} - 770 T^{3} + 32534 T^{4} + 66025 T^{5} - 3312205 T^{6} - 3580430 T^{7} + 267548612 T^{8} + 86879620 T^{9} - 17862005833 T^{10} + 86879620 p T^{11} + 267548612 p^{2} T^{12} - 3580430 p^{3} T^{13} - 3312205 p^{4} T^{14} + 66025 p^{5} T^{15} + 32534 p^{6} T^{16} - 770 p^{7} T^{17} - 233 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20}
67 1−16T+52T2+292T3−5179T4+79117T5−447406T6−998192T7+21597155T8−259219709T9+2885084279T10−259219709pT11+21597155p2T12−998192p3T13−447406p4T14+79117p5T15−5179p6T16+292p7T17+52p8T18−16p9T19+p10T20 1 - 16 T + 52 T^{2} + 292 T^{3} - 5179 T^{4} + 79117 T^{5} - 447406 T^{6} - 998192 T^{7} + 21597155 T^{8} - 259219709 T^{9} + 2885084279 T^{10} - 259219709 p T^{11} + 21597155 p^{2} T^{12} - 998192 p^{3} T^{13} - 447406 p^{4} T^{14} + 79117 p^{5} T^{15} - 5179 p^{6} T^{16} + 292 p^{7} T^{17} + 52 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20}
71 1+19T+13T2−1414T3−468T4+881pT5−611944T6−7048209T7+20077850T8+215658713T9−492609879T10+215658713pT11+20077850p2T12−7048209p3T13−611944p4T14+881p6T15−468p6T16−1414p7T17+13p8T18+19p9T19+p10T20 1 + 19 T + 13 T^{2} - 1414 T^{3} - 468 T^{4} + 881 p T^{5} - 611944 T^{6} - 7048209 T^{7} + 20077850 T^{8} + 215658713 T^{9} - 492609879 T^{10} + 215658713 p T^{11} + 20077850 p^{2} T^{12} - 7048209 p^{3} T^{13} - 611944 p^{4} T^{14} + 881 p^{6} T^{15} - 468 p^{6} T^{16} - 1414 p^{7} T^{17} + 13 p^{8} T^{18} + 19 p^{9} T^{19} + p^{10} T^{20}
73 (1+8T+246T2+1346T3+28129T4+121377T5+28129pT6+1346p2T7+246p3T8+8p4T9+p5T10)2 ( 1 + 8 T + 246 T^{2} + 1346 T^{3} + 28129 T^{4} + 121377 T^{5} + 28129 p T^{6} + 1346 p^{2} T^{7} + 246 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2}
79 (1+14T+300T2+3074T3+42145T4+327819T5+42145pT6+3074p2T7+300p3T8+14p4T9+p5T10)2 ( 1 + 14 T + 300 T^{2} + 3074 T^{3} + 42145 T^{4} + 327819 T^{5} + 42145 p T^{6} + 3074 p^{2} T^{7} + 300 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2}
83 (1+7T+158T2+1863T3+24007T4+159037T5+24007pT6+1863p2T7+158p3T8+7p4T9+p5T10)2 ( 1 + 7 T + 158 T^{2} + 1863 T^{3} + 24007 T^{4} + 159037 T^{5} + 24007 p T^{6} + 1863 p^{2} T^{7} + 158 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2}
89 1−10T−306T2+2112T3+65305T4−271977T5−10058402T6+21214470T7+1225818667T8−762210039T9−120541083185T10−762210039pT11+1225818667p2T12+21214470p3T13−10058402p4T14−271977p5T15+65305p6T16+2112p7T17−306p8T18−10p9T19+p10T20 1 - 10 T - 306 T^{2} + 2112 T^{3} + 65305 T^{4} - 271977 T^{5} - 10058402 T^{6} + 21214470 T^{7} + 1225818667 T^{8} - 762210039 T^{9} - 120541083185 T^{10} - 762210039 p T^{11} + 1225818667 p^{2} T^{12} + 21214470 p^{3} T^{13} - 10058402 p^{4} T^{14} - 271977 p^{5} T^{15} + 65305 p^{6} T^{16} + 2112 p^{7} T^{17} - 306 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20}
97 1+37T+516T2+4163T3+40703T4+282376T5−2037243T6−34573291T7−28158538T8−544608903T9−24070563673T10−544608903pT11−28158538p2T12−34573291p3T13−2037243p4T14+282376p5T15+40703p6T16+4163p7T17+516p8T18+37p9T19+p10T20 1 + 37 T + 516 T^{2} + 4163 T^{3} + 40703 T^{4} + 282376 T^{5} - 2037243 T^{6} - 34573291 T^{7} - 28158538 T^{8} - 544608903 T^{9} - 24070563673 T^{10} - 544608903 p T^{11} - 28158538 p^{2} T^{12} - 34573291 p^{3} T^{13} - 2037243 p^{4} T^{14} + 282376 p^{5} T^{15} + 40703 p^{6} T^{16} + 4163 p^{7} T^{17} + 516 p^{8} T^{18} + 37 p^{9} T^{19} + p^{10} T^{20}
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   L(s)=∏p ∏j=120(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.38253418394432217872515640317, −4.35070330811764194893977131323, −4.28453488737234209050662763348, −4.08601168261360109043459507197, −4.03809302786921102858973721705, −3.87922728683128571548391341225, −3.79963793488928692050183580793, −3.69305772236395202280858204778, −3.38437080392205097488660864211, −3.30855400539417621871621452516, −3.21827616564514736408726341938, −2.92536223700126030693975628002, −2.88896443726096609350629779134, −2.85734023305156338157693619844, −2.60844358081501628651745976562, −2.06890462838833844850198754787, −2.04691440609770714909605607105, −1.95445381155498410011126100140, −1.73230534991772285646266140660, −1.69373797949955124014676577145, −1.46033955569582225577643723300, −1.39726418800397312464984022544, −1.22249992441489060596500792987, −0.68477636603985020595255067532, −0.36304622185617679769022233344, 0.36304622185617679769022233344, 0.68477636603985020595255067532, 1.22249992441489060596500792987, 1.39726418800397312464984022544, 1.46033955569582225577643723300, 1.69373797949955124014676577145, 1.73230534991772285646266140660, 1.95445381155498410011126100140, 2.04691440609770714909605607105, 2.06890462838833844850198754787, 2.60844358081501628651745976562, 2.85734023305156338157693619844, 2.88896443726096609350629779134, 2.92536223700126030693975628002, 3.21827616564514736408726341938, 3.30855400539417621871621452516, 3.38437080392205097488660864211, 3.69305772236395202280858204778, 3.79963793488928692050183580793, 3.87922728683128571548391341225, 4.03809302786921102858973721705, 4.08601168261360109043459507197, 4.28453488737234209050662763348, 4.35070330811764194893977131323, 4.38253418394432217872515640317

Graph of the ZZ-function along the critical line

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