# Properties

 Degree $12$ Conductor $3.075\times 10^{13}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s − 3-s + 8·4-s − 4·6-s + 10·8-s − 2·9-s − 20·11-s − 8·12-s − 8·18-s + 4·19-s − 80·22-s − 18·23-s − 10·24-s + 13·25-s + 3·27-s − 28·32-s + 20·33-s − 16·36-s + 16·38-s − 160·44-s − 72·46-s + 46·47-s − 34·49-s + 52·50-s + 12·54-s − 4·57-s − 10·59-s + ⋯
 L(s)  = 1 + 2.82·2-s − 0.577·3-s + 4·4-s − 1.63·6-s + 3.53·8-s − 2/3·9-s − 6.03·11-s − 2.30·12-s − 1.88·18-s + 0.917·19-s − 17.0·22-s − 3.75·23-s − 2.04·24-s + 13/5·25-s + 0.577·27-s − 4.94·32-s + 3.48·33-s − 8/3·36-s + 2.59·38-s − 24.1·44-s − 10.6·46-s + 6.70·47-s − 4.85·49-s + 7.35·50-s + 1.63·54-s − 0.529·57-s − 1.30·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\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 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$3^{6} \cdot 59^{6}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{177} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 3^{6} \cdot 59^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.693946$$ $$L(\frac12)$$ $$\approx$$ $$0.693946$$ $$L(\frac{3}{2})$$ 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 + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6}$$
59 $$1 + 10 T + 85 T^{2} + 732 T^{3} + 85 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}$$
good2 $$( 1 - p T + p T^{2} - T^{3} + p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2}$$
5 $$1 - 13 T^{2} + 106 T^{4} - 637 T^{6} + 106 p^{2} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12}$$
7 $$( 1 + 17 T^{2} + T^{3} + 17 p T^{4} + p^{3} T^{6} )^{2}$$
11 $$( 1 + 10 T + 60 T^{2} + 234 T^{3} + 60 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
13 $$1 - 20 T^{2} + 516 T^{4} - 6542 T^{6} + 516 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12}$$
17 $$1 - 75 T^{2} + 2696 T^{4} - 57772 T^{6} + 2696 p^{2} T^{8} - 75 p^{4} T^{10} + p^{6} T^{12}$$
19 $$( 1 - 2 T + 31 T^{2} - 113 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
23 $$( 1 + 9 T + 4 p T^{2} + 428 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
29 $$1 - 120 T^{2} + 6965 T^{4} - 249781 T^{6} + 6965 p^{2} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12}$$
31 $$1 - 71 T^{2} + 3744 T^{4} - 137216 T^{6} + 3744 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12}$$
37 $$1 + T^{2} + 1068 T^{4} + 37816 T^{6} + 1068 p^{2} T^{8} + p^{4} T^{10} + p^{6} T^{12}$$
41 $$1 - 188 T^{2} + 15985 T^{4} - 816533 T^{6} + 15985 p^{2} T^{8} - 188 p^{4} T^{10} + p^{6} T^{12}$$
43 $$1 - 52 T^{2} + 5868 T^{4} - 182854 T^{6} + 5868 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12}$$
47 $$( 1 - 23 T + 312 T^{2} - 2568 T^{3} + 312 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
53 $$1 - 197 T^{2} + 19498 T^{4} - 1253909 T^{6} + 19498 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12}$$
61 $$1 - 167 T^{2} + 17204 T^{4} - 1305528 T^{6} + 17204 p^{2} T^{8} - 167 p^{4} T^{10} + p^{6} T^{12}$$
67 $$1 - 344 T^{2} + 52788 T^{4} - 4582178 T^{6} + 52788 p^{2} T^{8} - 344 p^{4} T^{10} + p^{6} T^{12}$$
71 $$1 - 120 T^{2} + 132 T^{4} + 542342 T^{6} + 132 p^{2} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12}$$
73 $$1 - 191 T^{2} + 21072 T^{4} - 1746740 T^{6} + 21072 p^{2} T^{8} - 191 p^{4} T^{10} + p^{6} T^{12}$$
79 $$( 1 + 7 T + 152 T^{2} + 759 T^{3} + 152 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$( 1 + 25 T + 400 T^{2} + 4164 T^{3} + 400 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
89 $$( 1 - 13 T + 242 T^{2} - 1796 T^{3} + 242 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
97 $$1 - 240 T^{2} + 15788 T^{4} - 353866 T^{6} + 15788 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$