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Properties

Label 12-91e12-1.1-c1e6-0-7
Degree 1212
Conductor 3.225×10233.225\times 10^{23}
Sign 11
Analytic cond. 8.35909×10108.35909\times 10^{10}
Root an. cond. 8.131678.13167
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 66

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 4·4-s − 6·5-s + 26·9-s − 4·11-s − 32·12-s − 48·15-s + 6·16-s + 16·17-s − 2·19-s + 24·20-s − 6·23-s + 25-s + 36·27-s − 6·29-s − 6·31-s + 4·32-s − 32·33-s − 104·36-s + 8·41-s + 2·43-s + 16·44-s − 156·45-s − 30·47-s + 48·48-s + 128·51-s − 14·53-s + ⋯
L(s)  = 1  + 4.61·3-s − 2·4-s − 2.68·5-s + 26/3·9-s − 1.20·11-s − 9.23·12-s − 12.3·15-s + 3/2·16-s + 3.88·17-s − 0.458·19-s + 5.36·20-s − 1.25·23-s + 1/5·25-s + 6.92·27-s − 1.11·29-s − 1.07·31-s + 0.707·32-s − 5.57·33-s − 17.3·36-s + 1.24·41-s + 0.304·43-s + 2.41·44-s − 23.2·45-s − 4.37·47-s + 6.92·48-s + 17.9·51-s − 1.92·53-s + ⋯

Functional equation

Λ(s)=((712⋅1312)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((712⋅1312)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 712⋅13127^{12} \cdot 13^{12}
Sign: 11
Analytic conductor: 8.35909×10108.35909\times 10^{10}
Root analytic conductor: 8.131678.13167
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 66
Selberg data: (12, 712⋅1312, ( :[1/2]6), 1)(12,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad7 1 1
13 1 1
good2 1+p2T2+5pT4−p2T5+11pT6−p3T7+5p3T8+p6T10+p6T12 1 + p^{2} T^{2} + 5 p T^{4} - p^{2} T^{5} + 11 p T^{6} - p^{3} T^{7} + 5 p^{3} T^{8} + p^{6} T^{10} + p^{6} T^{12}
3 1−8T+38T2−44pT3+121pT4−820T5+1550T6−820pT7+121p3T8−44p4T9+38p4T10−8p5T11+p6T12 1 - 8 T + 38 T^{2} - 44 p T^{3} + 121 p T^{4} - 820 T^{5} + 1550 T^{6} - 820 p T^{7} + 121 p^{3} T^{8} - 44 p^{4} T^{9} + 38 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
5 1+6T+7pT2+126T3+444T4+1166T5+2971T6+1166pT7+444p2T8+126p3T9+7p5T10+6p5T11+p6T12 1 + 6 T + 7 p T^{2} + 126 T^{3} + 444 T^{4} + 1166 T^{5} + 2971 T^{6} + 1166 p T^{7} + 444 p^{2} T^{8} + 126 p^{3} T^{9} + 7 p^{5} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
11 1+4T+28T2+64T3+329T4+384T5+2562T6+384pT7+329p2T8+64p3T9+28p4T10+4p5T11+p6T12 1 + 4 T + 28 T^{2} + 64 T^{3} + 329 T^{4} + 384 T^{5} + 2562 T^{6} + 384 p T^{7} + 329 p^{2} T^{8} + 64 p^{3} T^{9} + 28 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
17 1−16T+158T2−1036T3+5351T4−22924T5+95142T6−22924pT7+5351p2T8−1036p3T9+158p4T10−16p5T11+p6T12 1 - 16 T + 158 T^{2} - 1036 T^{3} + 5351 T^{4} - 22924 T^{5} + 95142 T^{6} - 22924 p T^{7} + 5351 p^{2} T^{8} - 1036 p^{3} T^{9} + 158 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}
19 1+2T+97T2+174T3+4206T4+6314T5+103439T6+6314pT7+4206p2T8+174p3T9+97p4T10+2p5T11+p6T12 1 + 2 T + 97 T^{2} + 174 T^{3} + 4206 T^{4} + 6314 T^{5} + 103439 T^{6} + 6314 p T^{7} + 4206 p^{2} T^{8} + 174 p^{3} T^{9} + 97 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
23 1+6T+101T2+418T3+4362T4+602pT5+5159pT6+602p2T7+4362p2T8+418p3T9+101p4T10+6p5T11+p6T12 1 + 6 T + 101 T^{2} + 418 T^{3} + 4362 T^{4} + 602 p T^{5} + 5159 p T^{6} + 602 p^{2} T^{7} + 4362 p^{2} T^{8} + 418 p^{3} T^{9} + 101 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
29 1+6T+141T2+602T3+8386T4+27374T5+298533T6+27374pT7+8386p2T8+602p3T9+141p4T10+6p5T11+p6T12 1 + 6 T + 141 T^{2} + 602 T^{3} + 8386 T^{4} + 27374 T^{5} + 298533 T^{6} + 27374 p T^{7} + 8386 p^{2} T^{8} + 602 p^{3} T^{9} + 141 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
31 1+6T+71T2+422T3+4336T4+20462T5+147703T6+20462pT7+4336p2T8+422p3T9+71p4T10+6p5T11+p6T12 1 + 6 T + 71 T^{2} + 422 T^{3} + 4336 T^{4} + 20462 T^{5} + 147703 T^{6} + 20462 p T^{7} + 4336 p^{2} T^{8} + 422 p^{3} T^{9} + 71 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
37 1+140T2−236T3+8777T4−25480T5+367738T6−25480pT7+8777p2T8−236p3T9+140p4T10+p6T12 1 + 140 T^{2} - 236 T^{3} + 8777 T^{4} - 25480 T^{5} + 367738 T^{6} - 25480 p T^{7} + 8777 p^{2} T^{8} - 236 p^{3} T^{9} + 140 p^{4} T^{10} + p^{6} T^{12}
41 1−8T+126T2−672T3+7255T4−35160T5+337924T6−35160pT7+7255p2T8−672p3T9+126p4T10−8p5T11+p6T12 1 - 8 T + 126 T^{2} - 672 T^{3} + 7255 T^{4} - 35160 T^{5} + 337924 T^{6} - 35160 p T^{7} + 7255 p^{2} T^{8} - 672 p^{3} T^{9} + 126 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
43 1−2T+97T2−618T3+6618T4−32018T5+404609T6−32018pT7+6618p2T8−618p3T9+97p4T10−2p5T11+p6T12 1 - 2 T + 97 T^{2} - 618 T^{3} + 6618 T^{4} - 32018 T^{5} + 404609 T^{6} - 32018 p T^{7} + 6618 p^{2} T^{8} - 618 p^{3} T^{9} + 97 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
47 1+30T+497T2+5570T3+47934T4+346670T5+2382079T6+346670pT7+47934p2T8+5570p3T9+497p4T10+30p5T11+p6T12 1 + 30 T + 497 T^{2} + 5570 T^{3} + 47934 T^{4} + 346670 T^{5} + 2382079 T^{6} + 346670 p T^{7} + 47934 p^{2} T^{8} + 5570 p^{3} T^{9} + 497 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12}
53 1+14T+281T2+3030T3+35490T4+288526T5+2479717T6+288526pT7+35490p2T8+3030p3T9+281p4T10+14p5T11+p6T12 1 + 14 T + 281 T^{2} + 3030 T^{3} + 35490 T^{4} + 288526 T^{5} + 2479717 T^{6} + 288526 p T^{7} + 35490 p^{2} T^{8} + 3030 p^{3} T^{9} + 281 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12}
59 1+24T+500T2+7036T3+85435T4+826892T5+7012620T6+826892pT7+85435p2T8+7036p3T9+500p4T10+24p5T11+p6T12 1 + 24 T + 500 T^{2} + 7036 T^{3} + 85435 T^{4} + 826892 T^{5} + 7012620 T^{6} + 826892 p T^{7} + 85435 p^{2} T^{8} + 7036 p^{3} T^{9} + 500 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12}
61 1+120T2+112T3+12043T4−272T5+813584T6−272pT7+12043p2T8+112p3T9+120p4T10+p6T12 1 + 120 T^{2} + 112 T^{3} + 12043 T^{4} - 272 T^{5} + 813584 T^{6} - 272 p T^{7} + 12043 p^{2} T^{8} + 112 p^{3} T^{9} + 120 p^{4} T^{10} + p^{6} T^{12}
67 1+16T+6pT2+4560T3+65351T4+561152T5+5755516T6+561152pT7+65351p2T8+4560p3T9+6p5T10+16p5T11+p6T12 1 + 16 T + 6 p T^{2} + 4560 T^{3} + 65351 T^{4} + 561152 T^{5} + 5755516 T^{6} + 561152 p T^{7} + 65351 p^{2} T^{8} + 4560 p^{3} T^{9} + 6 p^{5} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12}
71 1+8T+110T2+984T3+19145T4+111980T5+1119230T6+111980pT7+19145p2T8+984p3T9+110p4T10+8p5T11+p6T12 1 + 8 T + 110 T^{2} + 984 T^{3} + 19145 T^{4} + 111980 T^{5} + 1119230 T^{6} + 111980 p T^{7} + 19145 p^{2} T^{8} + 984 p^{3} T^{9} + 110 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
73 1−6T+197T2−1382T3+21574T4−165022T5+1685555T6−165022pT7+21574p2T8−1382p3T9+197p4T10−6p5T11+p6T12 1 - 6 T + 197 T^{2} - 1382 T^{3} + 21574 T^{4} - 165022 T^{5} + 1685555 T^{6} - 165022 p T^{7} + 21574 p^{2} T^{8} - 1382 p^{3} T^{9} + 197 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
79 1+22T+409T2+5518T3+61946T4+630742T5+5676321T6+630742pT7+61946p2T8+5518p3T9+409p4T10+22p5T11+p6T12 1 + 22 T + 409 T^{2} + 5518 T^{3} + 61946 T^{4} + 630742 T^{5} + 5676321 T^{6} + 630742 p T^{7} + 61946 p^{2} T^{8} + 5518 p^{3} T^{9} + 409 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12}
83 1+50T+1439T2+28742T3+5324pT4+5421474T5+54504063T6+5421474pT7+5324p3T8+28742p3T9+1439p4T10+50p5T11+p6T12 1 + 50 T + 1439 T^{2} + 28742 T^{3} + 5324 p T^{4} + 5421474 T^{5} + 54504063 T^{6} + 5421474 p T^{7} + 5324 p^{3} T^{8} + 28742 p^{3} T^{9} + 1439 p^{4} T^{10} + 50 p^{5} T^{11} + p^{6} T^{12}
89 1+26T+665T2+10382T3+156398T4+1742290T5+18723811T6+1742290pT7+156398p2T8+10382p3T9+665p4T10+26p5T11+p6T12 1 + 26 T + 665 T^{2} + 10382 T^{3} + 156398 T^{4} + 1742290 T^{5} + 18723811 T^{6} + 1742290 p T^{7} + 156398 p^{2} T^{8} + 10382 p^{3} T^{9} + 665 p^{4} T^{10} + 26 p^{5} T^{11} + p^{6} T^{12}
97 1−14T+321T2−3170T3+53038T4−457742T5+6291427T6−457742pT7+53038p2T8−3170p3T9+321p4T10−14p5T11+p6T12 1 - 14 T + 321 T^{2} - 3170 T^{3} + 53038 T^{4} - 457742 T^{5} + 6291427 T^{6} - 457742 p T^{7} + 53038 p^{2} T^{8} - 3170 p^{3} T^{9} + 321 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.36101440480386101698805551694, −3.95665515144764562722834729320, −3.90337791653529846972099510917, −3.87974589592575167622038733693, −3.77049180114536768141144197617, −3.74938636099151667818091477826, −3.59438862622294930673654970945, −3.33537730490894149019626877718, −3.24110744487946503093911134921, −3.08885952624629299592861918305, −3.05611024211759571866284408047, −3.04706319279418666703126649155, −2.85310698463864203488547050870, −2.82310046780077232960728487586, −2.69222049246310760348563700349, −2.33577796029993186202790757595, −2.26041625959125089347088584327, −2.22780521015848868957897687334, −1.87706057943070898978876178820, −1.58533235697894651083806206613, −1.54946450792298844460592020265, −1.54068681717765659733519646889, −1.22397620654511958981845069706, −1.21399960076514160510021024523, −0.861335467438693055237076129701, 0, 0, 0, 0, 0, 0, 0.861335467438693055237076129701, 1.21399960076514160510021024523, 1.22397620654511958981845069706, 1.54068681717765659733519646889, 1.54946450792298844460592020265, 1.58533235697894651083806206613, 1.87706057943070898978876178820, 2.22780521015848868957897687334, 2.26041625959125089347088584327, 2.33577796029993186202790757595, 2.69222049246310760348563700349, 2.82310046780077232960728487586, 2.85310698463864203488547050870, 3.04706319279418666703126649155, 3.05611024211759571866284408047, 3.08885952624629299592861918305, 3.24110744487946503093911134921, 3.33537730490894149019626877718, 3.59438862622294930673654970945, 3.74938636099151667818091477826, 3.77049180114536768141144197617, 3.87974589592575167622038733693, 3.90337791653529846972099510917, 3.95665515144764562722834729320, 4.36101440480386101698805551694

Graph of the ZZ-function along the critical line

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