Properties

Label 12-912e6-1.1-c1e6-0-2
Degree $12$
Conductor $5.754\times 10^{17}$
Sign $1$
Analytic cond. $149152.$
Root an. cond. $2.69858$
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  + 3·3-s − 2·5-s + 2·7-s + 3·9-s + 13-s − 6·15-s − 4·19-s + 6·21-s + 14·23-s + 7·25-s − 2·27-s − 4·29-s − 30·31-s − 4·35-s − 6·37-s + 3·39-s + 4·41-s − 3·43-s − 6·45-s − 18·47-s − 21·49-s + 6·53-s − 12·57-s − 13·61-s + 6·63-s − 2·65-s + 9·67-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s + 0.755·7-s + 9-s + 0.277·13-s − 1.54·15-s − 0.917·19-s + 1.30·21-s + 2.91·23-s + 7/5·25-s − 0.384·27-s − 0.742·29-s − 5.38·31-s − 0.676·35-s − 0.986·37-s + 0.480·39-s + 0.624·41-s − 0.457·43-s − 0.894·45-s − 2.62·47-s − 3·49-s + 0.824·53-s − 1.58·57-s − 1.66·61-s + 0.755·63-s − 0.248·65-s + 1.09·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(149152.\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{912} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.929191278\)
\(L(\frac12)\) \(\approx\) \(1.929191278\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - T + T^{2} )^{3} \)
19 \( 1 + 4 T + 17 T^{2} + 136 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 + 2 T - 3 T^{2} - 2 T^{3} - 2 T^{4} - 34 T^{5} - 31 T^{6} - 34 p T^{7} - 2 p^{2} T^{8} - 2 p^{3} T^{9} - 3 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( ( 1 - T + 12 T^{2} - 17 T^{3} + 12 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 + 9 T^{2} + 36 T^{3} + 9 p T^{4} + p^{3} T^{6} )^{2} \)
13 \( 1 - T - 17 T^{2} + 40 T^{3} + 61 T^{4} - 223 T^{5} + 854 T^{6} - 223 p T^{7} + 61 p^{2} T^{8} + 40 p^{3} T^{9} - 17 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
17 \( ( 1 - p T^{2} + p^{2} T^{4} )^{3} \)
23 \( 1 - 14 T + 99 T^{2} - 382 T^{3} + 346 T^{4} + 7726 T^{5} - 57613 T^{6} + 7726 p T^{7} + 346 p^{2} T^{8} - 382 p^{3} T^{9} + 99 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 4 T - 39 T^{2} - 52 T^{3} + 886 T^{4} - 1916 T^{5} - 33907 T^{6} - 1916 p T^{7} + 886 p^{2} T^{8} - 52 p^{3} T^{9} - 39 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( ( 1 + 15 T + 144 T^{2} + 29 p T^{3} + 144 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + T + p T^{2} )^{6} \)
41 \( 1 - 4 T - 75 T^{2} + 100 T^{3} + 3622 T^{4} + 2012 T^{5} - 174751 T^{6} + 2012 p T^{7} + 3622 p^{2} T^{8} + 100 p^{3} T^{9} - 75 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 3 T - 99 T^{2} - 218 T^{3} + 6207 T^{4} + 6951 T^{5} - 285738 T^{6} + 6951 p T^{7} + 6207 p^{2} T^{8} - 218 p^{3} T^{9} - 99 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{3} \)
53 \( 1 - 6 T - 51 T^{2} + 102 T^{3} + 1086 T^{4} + 11334 T^{5} - 107183 T^{6} + 11334 p T^{7} + 1086 p^{2} T^{8} + 102 p^{3} T^{9} - 51 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 153 T^{2} - 72 T^{3} + 14382 T^{4} + 5508 T^{5} - 969077 T^{6} + 5508 p T^{7} + 14382 p^{2} T^{8} - 72 p^{3} T^{9} - 153 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 + 13 T - 25 T^{2} - 504 T^{3} + 6133 T^{4} + 34211 T^{5} - 181514 T^{6} + 34211 p T^{7} + 6133 p^{2} T^{8} - 504 p^{3} T^{9} - 25 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 9 T - 39 T^{2} + 250 T^{3} + 375 T^{4} + 24519 T^{5} - 284658 T^{6} + 24519 p T^{7} + 375 p^{2} T^{8} + 250 p^{3} T^{9} - 39 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 18 T + 99 T^{2} + 234 T^{3} + 306 T^{4} - 55062 T^{5} - 871373 T^{6} - 55062 p T^{7} + 306 p^{2} T^{8} + 234 p^{3} T^{9} + 99 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 19 T + 59 T^{2} + 252 T^{3} + 17041 T^{4} + 91889 T^{5} - 279578 T^{6} + 91889 p T^{7} + 17041 p^{2} T^{8} + 252 p^{3} T^{9} + 59 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 11 T - 119 T^{2} + 494 T^{3} + 21403 T^{4} - 27611 T^{5} - 1863994 T^{6} - 27611 p T^{7} + 21403 p^{2} T^{8} + 494 p^{3} T^{9} - 119 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 4 T + 169 T^{2} - 472 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 16 T - 87 T^{2} + 424 T^{3} + 39826 T^{4} - 163780 T^{5} - 2350663 T^{6} - 163780 p T^{7} + 39826 p^{2} T^{8} + 424 p^{3} T^{9} - 87 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 2 T - 19 T^{2} - 2166 T^{3} + 418 T^{4} + 14486 T^{5} + 2859997 T^{6} + 14486 p T^{7} + 418 p^{2} T^{8} - 2166 p^{3} T^{9} - 19 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.18091671986280360337340663577, −5.10059652804735857263724118839, −4.94156578610376667125273716063, −4.82035851860667695339351919157, −4.80860422408671284565946724250, −4.78496160871144732162276814711, −4.10572417762547674284809227890, −4.05288195739382967816771783182, −4.01358151081215318121851385750, −3.71050838011052802892397165134, −3.61896890045984704043603794984, −3.37679528059963927372121302262, −3.37616086036901105644933970646, −3.20418563321536068065469943683, −2.88744657702586208518653963793, −2.70808361790408769566856713064, −2.60874542782096559274396435377, −2.41266216664206534659795890326, −1.87335476968152249903907577889, −1.70398043451543250361502145971, −1.64158599577505187000417045552, −1.58567693003855994962986159818, −1.31163375389445785080215730663, −0.55164722450051100403758847716, −0.22316160599425444163387351790, 0.22316160599425444163387351790, 0.55164722450051100403758847716, 1.31163375389445785080215730663, 1.58567693003855994962986159818, 1.64158599577505187000417045552, 1.70398043451543250361502145971, 1.87335476968152249903907577889, 2.41266216664206534659795890326, 2.60874542782096559274396435377, 2.70808361790408769566856713064, 2.88744657702586208518653963793, 3.20418563321536068065469943683, 3.37616086036901105644933970646, 3.37679528059963927372121302262, 3.61896890045984704043603794984, 3.71050838011052802892397165134, 4.01358151081215318121851385750, 4.05288195739382967816771783182, 4.10572417762547674284809227890, 4.78496160871144732162276814711, 4.80860422408671284565946724250, 4.82035851860667695339351919157, 4.94156578610376667125273716063, 5.10059652804735857263724118839, 5.18091671986280360337340663577

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

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