# 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 of factors

## 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.