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Properties

Label 12-91e12-1.1-c1e6-0-7
Degree $12$
Conductor $3.225\times 10^{23}$
Sign $1$
Analytic cond. $8.35909\times 10^{10}$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(8.35909\times 10^{10}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line

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