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Properties

Label 12-5415e6-1.1-c1e6-0-8
Degree $12$
Conductor $2.521\times 10^{22}$
Sign $1$
Analytic cond. $6.53511\times 10^{9}$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 6·3-s + 2·4-s − 6·5-s − 18·6-s + 6·7-s + 8-s + 21·9-s + 18·10-s − 4·11-s + 12·12-s − 8·13-s − 18·14-s − 36·15-s − 16-s − 11·17-s − 63·18-s − 12·20-s + 36·21-s + 12·22-s − 3·23-s + 6·24-s + 21·25-s + 24·26-s + 56·27-s + 12·28-s − 7·29-s + ⋯
L(s)  = 1  − 2.12·2-s + 3.46·3-s + 4-s − 2.68·5-s − 7.34·6-s + 2.26·7-s + 0.353·8-s + 7·9-s + 5.69·10-s − 1.20·11-s + 3.46·12-s − 2.21·13-s − 4.81·14-s − 9.29·15-s − 1/4·16-s − 2.66·17-s − 14.8·18-s − 2.68·20-s + 7.85·21-s + 2.55·22-s − 0.625·23-s + 1.22·24-s + 21/5·25-s + 4.70·26-s + 10.7·27-s + 2.26·28-s − 1.29·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(6.53511\times 10^{9}\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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)$
bad3 \( ( 1 - T )^{6} \)
5 \( ( 1 + T )^{6} \)
19 \( 1 \)
good2 \( 1 + 3 T + 7 T^{2} + 7 p T^{3} + 13 p T^{4} + 45 T^{5} + 65 T^{6} + 45 p T^{7} + 13 p^{3} T^{8} + 7 p^{4} T^{9} + 7 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 6 T + 6 p T^{2} - 23 p T^{3} + 663 T^{4} - 1885 T^{5} + 5853 T^{6} - 1885 p T^{7} + 663 p^{2} T^{8} - 23 p^{4} T^{9} + 6 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 4 T + 49 T^{2} + 185 T^{3} + 105 p T^{4} + 3734 T^{5} + 16105 T^{6} + 3734 p T^{7} + 105 p^{3} T^{8} + 185 p^{3} T^{9} + 49 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 8 T + 63 T^{2} + 296 T^{3} + 1414 T^{4} + 5064 T^{5} + 20219 T^{6} + 5064 p T^{7} + 1414 p^{2} T^{8} + 296 p^{3} T^{9} + 63 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 11 T + 126 T^{2} + 846 T^{3} + 330 p T^{4} + 26921 T^{5} + 127667 T^{6} + 26921 p T^{7} + 330 p^{3} T^{8} + 846 p^{3} T^{9} + 126 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 3 T + 74 T^{2} + 152 T^{3} + 2380 T^{4} + 3673 T^{5} + 55763 T^{6} + 3673 p T^{7} + 2380 p^{2} T^{8} + 152 p^{3} T^{9} + 74 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 7 T + 121 T^{2} + 547 T^{3} + 6002 T^{4} + 19622 T^{5} + 193091 T^{6} + 19622 p T^{7} + 6002 p^{2} T^{8} + 547 p^{3} T^{9} + 121 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - T + 66 T^{2} - 108 T^{3} + 2334 T^{4} - 10109 T^{5} + 76533 T^{6} - 10109 p T^{7} + 2334 p^{2} T^{8} - 108 p^{3} T^{9} + 66 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 5 T + 124 T^{2} - 311 T^{3} + 5607 T^{4} - 1212 T^{5} + 177221 T^{6} - 1212 p T^{7} + 5607 p^{2} T^{8} - 311 p^{3} T^{9} + 124 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 19 T + 256 T^{2} + 2367 T^{3} + 19049 T^{4} + 134426 T^{5} + 896183 T^{6} + 134426 p T^{7} + 19049 p^{2} T^{8} + 2367 p^{3} T^{9} + 256 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 8 T + 137 T^{2} - 1038 T^{3} + 11143 T^{4} - 73266 T^{5} + 563755 T^{6} - 73266 p T^{7} + 11143 p^{2} T^{8} - 1038 p^{3} T^{9} + 137 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 4 T + 234 T^{2} + 818 T^{3} + 24541 T^{4} + 72311 T^{5} + 1481239 T^{6} + 72311 p T^{7} + 24541 p^{2} T^{8} + 818 p^{3} T^{9} + 234 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 24 T + 514 T^{2} + 6898 T^{3} + 83286 T^{4} + 756986 T^{5} + 6238987 T^{6} + 756986 p T^{7} + 83286 p^{2} T^{8} + 6898 p^{3} T^{9} + 514 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 25 T + 419 T^{2} + 5169 T^{3} + 53110 T^{4} + 481858 T^{5} + 3865139 T^{6} + 481858 p T^{7} + 53110 p^{2} T^{8} + 5169 p^{3} T^{9} + 419 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 8 T + 321 T^{2} - 2328 T^{3} + 45102 T^{4} - 277763 T^{5} + 3567655 T^{6} - 277763 p T^{7} + 45102 p^{2} T^{8} - 2328 p^{3} T^{9} + 321 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 17 T + 429 T^{2} + 4565 T^{3} + 65444 T^{4} + 508532 T^{5} + 5489699 T^{6} + 508532 p T^{7} + 65444 p^{2} T^{8} + 4565 p^{3} T^{9} + 429 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 5 T + 36 T^{2} - 252 T^{3} + 12965 T^{4} - 41876 T^{5} + 370681 T^{6} - 41876 p T^{7} + 12965 p^{2} T^{8} - 252 p^{3} T^{9} + 36 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 30 T + 635 T^{2} - 9056 T^{3} + 108810 T^{4} - 1069415 T^{5} + 9754773 T^{6} - 1069415 p T^{7} + 108810 p^{2} T^{8} - 9056 p^{3} T^{9} + 635 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 11 T + 148 T^{2} - 1685 T^{3} + 20589 T^{4} - 171090 T^{5} + 1800329 T^{6} - 171090 p T^{7} + 20589 p^{2} T^{8} - 1685 p^{3} T^{9} + 148 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 35 T + 772 T^{2} + 12595 T^{3} + 166247 T^{4} + 1850170 T^{5} + 18050265 T^{6} + 1850170 p T^{7} + 166247 p^{2} T^{8} + 12595 p^{3} T^{9} + 772 p^{4} T^{10} + 35 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + T + 289 T^{2} + 1453 T^{3} + 37414 T^{4} + 332196 T^{5} + 3439897 T^{6} + 332196 p T^{7} + 37414 p^{2} T^{8} + 1453 p^{3} T^{9} + 289 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 35 T + 910 T^{2} + 16842 T^{3} + 260090 T^{4} + 3264355 T^{5} + 35165117 T^{6} + 3264355 p T^{7} + 260090 p^{2} T^{8} + 16842 p^{3} T^{9} + 910 p^{4} T^{10} + 35 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.50323933245401191427652606989, −4.44932976226109902877930777896, −4.21463659857322065746949847544, −4.19188914743228852270631521911, −3.98477399656446447527198402690, −3.90346151747974274628153997205, −3.71995208016004054012901161203, −3.53242804436886728463793929057, −3.52282969849027823845365476581, −3.28537781170029584162472215648, −3.17250488166235387833157622654, −2.91515055109669332868699932268, −2.89403686500773214160712326130, −2.64523055634043573779349953148, −2.64245497410107317018738807901, −2.44733663277024804139203609344, −2.27927477176161048755361728202, −2.06938390115333278408192514601, −2.05882051094608685596655100523, −1.81158678351963290332499651754, −1.71562696601697650196301630384, −1.22964350845240446088678710037, −1.22208236859739295016405010759, −1.21973776513287195525629358095, −1.15417992666091405754846480581, 0, 0, 0, 0, 0, 0, 1.15417992666091405754846480581, 1.21973776513287195525629358095, 1.22208236859739295016405010759, 1.22964350845240446088678710037, 1.71562696601697650196301630384, 1.81158678351963290332499651754, 2.05882051094608685596655100523, 2.06938390115333278408192514601, 2.27927477176161048755361728202, 2.44733663277024804139203609344, 2.64245497410107317018738807901, 2.64523055634043573779349953148, 2.89403686500773214160712326130, 2.91515055109669332868699932268, 3.17250488166235387833157622654, 3.28537781170029584162472215648, 3.52282969849027823845365476581, 3.53242804436886728463793929057, 3.71995208016004054012901161203, 3.90346151747974274628153997205, 3.98477399656446447527198402690, 4.19188914743228852270631521911, 4.21463659857322065746949847544, 4.44932976226109902877930777896, 4.50323933245401191427652606989

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

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