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Properties

Label 12-637e6-1.1-c1e6-0-4
Degree $12$
Conductor $6.681\times 10^{16}$
Sign $1$
Analytic cond. $17318.0$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

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 + 6·13-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 + 48·39-s − 8·41-s + 2·43-s − 16·44-s + 156·45-s + 30·47-s + 48·48-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 + 1.66·13-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 + 7.68·39-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 + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(17318.0\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(33.83686847\)
\(L(\frac12)\) \(\approx\) \(33.83686847\)
\(L(\frac{3}{2})\) 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 \( ( 1 - T )^{6} \)
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

−5.64094557731621943140048115827, −5.61686331131908083242876772755, −5.26562698114258209798116913137, −5.21774013402822909505358741877, −5.07536770607284294125650802195, −4.84002741993096800399995300541, −4.58346017908502434123911103175, −4.01975644818778433237683685396, −3.88347837931537938364368641706, −3.83667107048168242498577795091, −3.78307621201181422753251217693, −3.75545087961897204495740710977, −3.70465499603267629391003055171, −3.24715820190858717330829287400, −3.09389315517282991663458065506, −2.84860105763173388855996623049, −2.81676841237292859568487250860, −2.35237178692968263429215304691, −2.25793862836781430891558143955, −2.19542371399468063645972123031, −1.82639530032567334447259075068, −1.68756983034729400615022091457, −1.42185787271355655763855117191, −0.907931743152917920890736717421, −0.77488385266128951688398560546, 0.77488385266128951688398560546, 0.907931743152917920890736717421, 1.42185787271355655763855117191, 1.68756983034729400615022091457, 1.82639530032567334447259075068, 2.19542371399468063645972123031, 2.25793862836781430891558143955, 2.35237178692968263429215304691, 2.81676841237292859568487250860, 2.84860105763173388855996623049, 3.09389315517282991663458065506, 3.24715820190858717330829287400, 3.70465499603267629391003055171, 3.75545087961897204495740710977, 3.78307621201181422753251217693, 3.83667107048168242498577795091, 3.88347837931537938364368641706, 4.01975644818778433237683685396, 4.58346017908502434123911103175, 4.84002741993096800399995300541, 5.07536770607284294125650802195, 5.21774013402822909505358741877, 5.26562698114258209798116913137, 5.61686331131908083242876772755, 5.64094557731621943140048115827

Graph of the $Z$-function along the critical line

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