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Properties

Label 12-931e6-1.1-c1e6-0-14
Degree $12$
Conductor $6.512\times 10^{17}$
Sign $1$
Analytic cond. $168795.$
Root an. cond. $2.72654$
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 − 3·3-s + 9·4-s + 3·5-s − 9·6-s + 18·8-s − 3·9-s + 9·10-s − 27·12-s − 3·13-s − 9·15-s + 36·16-s + 3·17-s − 9·18-s + 6·19-s + 27·20-s + 6·23-s − 54·24-s + 9·25-s − 9·26-s + 17·27-s − 3·29-s − 27·30-s + 9·31-s + 66·32-s + 9·34-s − 27·36-s + ⋯
L(s)  = 1  + 2.12·2-s − 1.73·3-s + 9/2·4-s + 1.34·5-s − 3.67·6-s + 6.36·8-s − 9-s + 2.84·10-s − 7.79·12-s − 0.832·13-s − 2.32·15-s + 9·16-s + 0.727·17-s − 2.12·18-s + 1.37·19-s + 6.03·20-s + 1.25·23-s − 11.0·24-s + 9/5·25-s − 1.76·26-s + 3.27·27-s − 0.557·29-s − 4.92·30-s + 1.61·31-s + 11.6·32-s + 1.54·34-s − 9/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(168795.\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(29.17140293\)
\(L(\frac12)\) \(\approx\) \(29.17140293\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 - 6 T + 42 T^{2} - 155 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
good2 \( 1 - 3 T + 9 T^{3} - 9 T^{4} - 3 p^{2} T^{5} + 37 T^{6} - 3 p^{3} T^{7} - 9 p^{2} T^{8} + 9 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
3 \( 1 + p T + 4 p T^{2} + 28 T^{3} + 26 p T^{4} + 47 p T^{5} + 289 T^{6} + 47 p^{2} T^{7} + 26 p^{3} T^{8} + 28 p^{3} T^{9} + 4 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T + 18 T^{3} - 36 T^{4} - 3 p^{2} T^{5} + 379 T^{6} - 3 p^{3} T^{7} - 36 p^{2} T^{8} + 18 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12} \)
11 \( ( 1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6} )^{2} \)
13 \( 1 + 3 T + 24 T^{2} + 2 p T^{3} + 315 T^{4} + 261 T^{5} + 4905 T^{6} + 261 p T^{7} + 315 p^{2} T^{8} + 2 p^{4} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 3 T + 18 T^{2} - 126 T^{3} + 621 T^{4} - 2217 T^{5} + 10549 T^{6} - 2217 p T^{7} + 621 p^{2} T^{8} - 126 p^{3} T^{9} + 18 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 162 T^{3} - 684 T^{4} - 6 p^{2} T^{5} + 32293 T^{6} - 6 p^{3} T^{7} - 684 p^{2} T^{8} + 162 p^{3} T^{9} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 3 T + 36 T^{2} + 378 T^{3} + 1872 T^{4} + 9921 T^{5} + 94159 T^{6} + 9921 p T^{7} + 1872 p^{2} T^{8} + 378 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 9 T - 18 T^{2} + 119 T^{3} + 2187 T^{4} - 3402 T^{5} - 67065 T^{6} - 3402 p T^{7} + 2187 p^{2} T^{8} + 119 p^{3} T^{9} - 18 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 90 T^{2} - 34 T^{3} + 4770 T^{4} + 1530 T^{5} - 198105 T^{6} + 1530 p T^{7} + 4770 p^{2} T^{8} - 34 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 21 T + 162 T^{2} - 180 T^{3} - 4707 T^{4} + 28401 T^{5} - 103463 T^{6} + 28401 p T^{7} - 4707 p^{2} T^{8} - 180 p^{3} T^{9} + 162 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T + 60 T^{2} + 8 T^{3} + 2691 T^{4} + 2799 T^{5} + 147141 T^{6} + 2799 p T^{7} + 2691 p^{2} T^{8} + 8 p^{3} T^{9} + 60 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 15 T + 72 T^{2} + 198 T^{3} - 3321 T^{4} + 7869 T^{5} + 15049 T^{6} + 7869 p T^{7} - 3321 p^{2} T^{8} + 198 p^{3} T^{9} + 72 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 15 T + 90 T^{2} + 18 T^{3} - 3447 T^{4} + 44751 T^{5} - 378575 T^{6} + 44751 p T^{7} - 3447 p^{2} T^{8} + 18 p^{3} T^{9} + 90 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T - 18 T^{2} + 540 T^{3} - 180 T^{4} - 5160 T^{5} + 467533 T^{6} - 5160 p T^{7} - 180 p^{2} T^{8} + 540 p^{3} T^{9} - 18 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 24 T + 276 T^{2} - 2314 T^{3} + 13356 T^{4} - 34164 T^{5} - 1521 T^{6} - 34164 p T^{7} + 13356 p^{2} T^{8} - 2314 p^{3} T^{9} + 276 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 24 T + 456 T^{2} - 6454 T^{3} + 78804 T^{4} - 795726 T^{5} + 7018533 T^{6} - 795726 p T^{7} + 78804 p^{2} T^{8} - 6454 p^{3} T^{9} + 456 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T - 36 T^{2} + 594 T^{3} + 3240 T^{4} + 14892 T^{5} + 665785 T^{6} + 14892 p T^{7} + 3240 p^{2} T^{8} + 594 p^{3} T^{9} - 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 24 T + 240 T^{2} - 1216 T^{3} - 432 T^{4} + 102924 T^{5} - 1320489 T^{6} + 102924 p T^{7} - 432 p^{2} T^{8} - 1216 p^{3} T^{9} + 240 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 15 T + 87 T^{2} + 179 T^{3} - 1674 T^{4} - 76626 T^{5} + 1419201 T^{6} - 76626 p T^{7} - 1674 p^{2} T^{8} + 179 p^{3} T^{9} + 87 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 60 T^{2} + 918 T^{3} - 1380 T^{4} - 27540 T^{5} + 1055455 T^{6} - 27540 p T^{7} - 1380 p^{2} T^{8} + 918 p^{3} T^{9} - 60 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 3 T + 99 T^{2} + 1251 T^{3} + 17118 T^{4} + 120972 T^{5} + 1657225 T^{6} + 120972 p T^{7} + 17118 p^{2} T^{8} + 1251 p^{3} T^{9} + 99 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 18 T + 234 T^{2} - 3310 T^{3} + 39204 T^{4} - 371520 T^{5} + 3748107 T^{6} - 371520 p T^{7} + 39204 p^{2} T^{8} - 3310 p^{3} T^{9} + 234 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.33513155798511619731332611959, −5.33126160822428658235672661328, −5.28420376581078181448373372460, −5.08805943876810696858105585511, −4.77199356730825145681315303433, −4.63279842906843976087495331209, −4.39952110015456288719523306531, −4.36355547332581799406137214644, −4.03669401063858421425938259252, −3.61256862871365227917880992398, −3.57896110957899088100490345680, −3.43858584966076556311096519681, −3.32279538663931539322575522255, −3.25647040399714265114750755846, −2.55742850845518141558566547189, −2.54703263670113480522243379379, −2.46212462301878721513365145561, −2.38295294554761517045444715893, −2.38067849486018991153373815827, −2.37392580631929982877874809466, −1.57200204403479580959510099856, −1.17231938208277706127000141134, −1.01045233274117805227602290997, −0.937857234262743668823727715256, −0.59177756046573222050693271225, 0.59177756046573222050693271225, 0.937857234262743668823727715256, 1.01045233274117805227602290997, 1.17231938208277706127000141134, 1.57200204403479580959510099856, 2.37392580631929982877874809466, 2.38067849486018991153373815827, 2.38295294554761517045444715893, 2.46212462301878721513365145561, 2.54703263670113480522243379379, 2.55742850845518141558566547189, 3.25647040399714265114750755846, 3.32279538663931539322575522255, 3.43858584966076556311096519681, 3.57896110957899088100490345680, 3.61256862871365227917880992398, 4.03669401063858421425938259252, 4.36355547332581799406137214644, 4.39952110015456288719523306531, 4.63279842906843976087495331209, 4.77199356730825145681315303433, 5.08805943876810696858105585511, 5.28420376581078181448373372460, 5.33126160822428658235672661328, 5.33513155798511619731332611959

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

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