Properties

Degree 12
Conductor $ 5^{6} \cdot 13^{6} \cdot 31^{6} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 6·5-s − 6-s − 7-s − 6·10-s + 11-s − 6·13-s + 14-s + 6·15-s + 17-s − 21-s − 22-s + 23-s + 21·25-s + 6·26-s − 6·30-s − 6·31-s + 33-s − 34-s − 6·35-s − 6·39-s + 42-s + 43-s − 46-s − 47-s − 21·50-s + ⋯
L(s)  = 1  − 2-s + 3-s + 6·5-s − 6-s − 7-s − 6·10-s + 11-s − 6·13-s + 14-s + 6·15-s + 17-s − 21-s − 22-s + 23-s + 21·25-s + 6·26-s − 6·30-s − 6·31-s + 33-s − 34-s − 6·35-s − 6·39-s + 42-s + 43-s − 46-s − 47-s − 21·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(5^{6} \cdot 13^{6} \cdot 31^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2015} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 5^{6} \cdot 13^{6} \cdot 31^{6} ,\ ( \ : [0]^{6} ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.850538789\)
\(L(\frac12)\)  \(\approx\)  \(1.850538789\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;13,\;31\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{5,\;13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad5 \( ( 1 - T )^{6} \)
13 \( ( 1 + T )^{6} \)
31 \( ( 1 + T )^{6} \)
good2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
3 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
11 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
17 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
19 \( ( 1 - T )^{6}( 1 + T )^{6} \)
23 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
29 \( ( 1 - T )^{6}( 1 + T )^{6} \)
37 \( ( 1 - T )^{6}( 1 + T )^{6} \)
41 \( ( 1 - T )^{6}( 1 + T )^{6} \)
43 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
47 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
53 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
59 \( ( 1 - T )^{6}( 1 + T )^{6} \)
61 \( ( 1 - T )^{6}( 1 + T )^{6} \)
67 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
71 \( ( 1 - T )^{6}( 1 + T )^{6} \)
73 \( ( 1 - T )^{6}( 1 + T )^{6} \)
79 \( ( 1 - T )^{6}( 1 + T )^{6} \)
83 \( ( 1 - T )^{6}( 1 + T )^{6} \)
89 \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
97 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.19236727407332585456642931927, −4.88061879317431361335496258371, −4.83317905439022467962790964152, −4.71595475025130215550375973528, −4.67279901674174423321318360159, −4.23955465756395753043709103724, −4.12318974184031417557261487514, −3.88693579701816528991417045468, −3.48256531394049166765026733114, −3.43831153160476014823201189282, −3.36335821906628261504053968807, −3.00319639382353361680295155149, −2.83489815514316088580991592171, −2.74722524968468503534655783056, −2.70983948529623984331796936912, −2.57045509104693024712292247872, −2.28209804667836791279486520083, −2.17478861680126184669907456665, −2.07112127568161902052360501432, −1.81549620389973541533802470866, −1.77262720499890806050144658160, −1.51671490132805965991072806610, −1.43062094492485354654680844975, −0.976066662750734410475717357659, −0.50326640238114147658436930821, 0.50326640238114147658436930821, 0.976066662750734410475717357659, 1.43062094492485354654680844975, 1.51671490132805965991072806610, 1.77262720499890806050144658160, 1.81549620389973541533802470866, 2.07112127568161902052360501432, 2.17478861680126184669907456665, 2.28209804667836791279486520083, 2.57045509104693024712292247872, 2.70983948529623984331796936912, 2.74722524968468503534655783056, 2.83489815514316088580991592171, 3.00319639382353361680295155149, 3.36335821906628261504053968807, 3.43831153160476014823201189282, 3.48256531394049166765026733114, 3.88693579701816528991417045468, 4.12318974184031417557261487514, 4.23955465756395753043709103724, 4.67279901674174423321318360159, 4.71595475025130215550375973528, 4.83317905439022467962790964152, 4.88061879317431361335496258371, 5.19236727407332585456642931927

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

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