Properties

Label 12-21e12-1.1-c1e6-0-10
Degree $12$
Conductor $7.356\times 10^{15}$
Sign $1$
Analytic cond. $1906.75$
Root an. cond. $1.87654$
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  − 3·2-s + 6·4-s + 6·5-s − 9·8-s − 18·10-s + 12·11-s + 3·13-s + 12·16-s − 6·17-s + 3·19-s + 36·20-s − 36·22-s + 24·23-s − 3·25-s − 9·26-s + 9·27-s − 9·29-s + 3·31-s − 12·32-s + 18·34-s + 3·37-s − 9·38-s − 54·40-s + 3·43-s + 72·44-s − 72·46-s − 3·47-s + ⋯
L(s)  = 1  − 2.12·2-s + 3·4-s + 2.68·5-s − 3.18·8-s − 5.69·10-s + 3.61·11-s + 0.832·13-s + 3·16-s − 1.45·17-s + 0.688·19-s + 8.04·20-s − 7.67·22-s + 5.00·23-s − 3/5·25-s − 1.76·26-s + 1.73·27-s − 1.67·29-s + 0.538·31-s − 2.12·32-s + 3.08·34-s + 0.493·37-s − 1.45·38-s − 8.53·40-s + 0.457·43-s + 10.8·44-s − 10.6·46-s − 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1906.75\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.925937984\)
\(L(\frac12)\) \(\approx\) \(3.925937984\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{2} T^{3} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( 1 + 3 T + 3 T^{2} - 3 T^{4} - 3 p T^{5} - 11 T^{6} - 3 p^{2} T^{7} - 3 p^{2} T^{8} + 3 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 - 3 T + 3 p T^{2} - 27 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 6 T + 42 T^{2} - 135 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 135 p T^{7} + 45 p^{2} T^{8} - 76 p^{3} T^{9} + 3 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 6 T - 24 T^{2} - 54 T^{3} + 1338 T^{4} + 1914 T^{5} - 18929 T^{6} + 1914 p T^{7} + 1338 p^{2} T^{8} - 54 p^{3} T^{9} - 24 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 42 T^{2} + 41 T^{3} + 1341 T^{4} - 216 T^{5} - 29541 T^{6} - 216 p T^{7} + 1341 p^{2} T^{8} + 41 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 - 12 T + 96 T^{2} - 549 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 9 T + 30 T^{2} + 81 T^{3} - 579 T^{4} - 9414 T^{5} - 59051 T^{6} - 9414 p T^{7} - 579 p^{2} T^{8} + 81 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T - 6 T^{2} - 319 T^{3} + 171 T^{4} + 1962 T^{5} + 62727 T^{6} + 1962 p T^{7} + 171 p^{2} T^{8} - 319 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 24 T^{2} - 301 T^{3} + 171 T^{4} + 6552 T^{5} + 58893 T^{6} + 6552 p T^{7} + 171 p^{2} T^{8} - 301 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 114 T^{2} - 18 T^{3} + 8322 T^{4} + 1026 T^{5} - 394913 T^{6} + 1026 p T^{7} + 8322 p^{2} T^{8} - 18 p^{3} T^{9} - 114 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 5670 p T^{7} + 9063 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 3 T - 78 T^{2} - 405 T^{3} + 2481 T^{4} + 11064 T^{5} - 57089 T^{6} + 11064 p T^{7} + 2481 p^{2} T^{8} - 405 p^{3} T^{9} - 78 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 6 T - 114 T^{2} - 378 T^{3} + 10716 T^{4} + 17304 T^{5} - 587549 T^{6} + 17304 p T^{7} + 10716 p^{2} T^{8} - 378 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 3 T - 96 T^{2} + 495 T^{3} + 3615 T^{4} - 15798 T^{5} - 107021 T^{6} - 15798 p T^{7} + 3615 p^{2} T^{8} + 495 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 19134 p T^{7} + 13698 p^{2} T^{8} - 418 p^{3} T^{9} - 132 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 50094 p T^{7} + 15318 p^{2} T^{8} + 518 p^{3} T^{9} - 78 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 21 T + 138 T^{2} - 769 T^{3} + 10953 T^{4} - 30402 T^{5} - 450903 T^{6} - 30402 p T^{7} + 10953 p^{2} T^{8} - 769 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 195678 p T^{7} + 25767 p^{2} T^{8} - 499 p^{3} T^{9} + 84 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 18 T + 30 T^{2} + 702 T^{3} + 8088 T^{4} - 126648 T^{5} + 719359 T^{6} - 126648 p T^{7} + 8088 p^{2} T^{8} + 702 p^{3} T^{9} + 30 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T - 60 T^{2} - 198 T^{3} + 7584 T^{4} - 70800 T^{5} - 1684181 T^{6} - 70800 p T^{7} + 7584 p^{2} T^{8} - 198 p^{3} T^{9} - 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 11502 p T^{7} + 2421 p^{2} T^{8} + 149 p^{3} T^{9} - 114 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−6.16955009059549070888603120061, −6.11140711525917726190799891190, −5.87021341142985628931316088727, −5.38436388529616352901938425534, −5.38230812697238564514380319898, −5.22217770478616139700750613429, −5.12023878697176692742483840359, −4.92223315931463880684090212539, −4.70079234571926891255859011501, −4.19676205139258560891287878343, −3.95165543361171249301758955541, −3.87056333820067728326078018899, −3.86459180547307648357347082800, −3.50526435110638626345219114257, −3.36873537750804530837581543237, −2.86044548557032563375394780558, −2.61316087231832447365979469722, −2.50513848106726247676398584109, −2.29588297420373919613456075033, −2.10457517115587742351704721888, −1.50792248904301098063192232970, −1.40000977447550774505957249211, −1.38781723698542096110292637509, −1.16988591322485282112455651317, −0.70848730091501298937955083459, 0.70848730091501298937955083459, 1.16988591322485282112455651317, 1.38781723698542096110292637509, 1.40000977447550774505957249211, 1.50792248904301098063192232970, 2.10457517115587742351704721888, 2.29588297420373919613456075033, 2.50513848106726247676398584109, 2.61316087231832447365979469722, 2.86044548557032563375394780558, 3.36873537750804530837581543237, 3.50526435110638626345219114257, 3.86459180547307648357347082800, 3.87056333820067728326078018899, 3.95165543361171249301758955541, 4.19676205139258560891287878343, 4.70079234571926891255859011501, 4.92223315931463880684090212539, 5.12023878697176692742483840359, 5.22217770478616139700750613429, 5.38230812697238564514380319898, 5.38436388529616352901938425534, 5.87021341142985628931316088727, 6.11140711525917726190799891190, 6.16955009059549070888603120061

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

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