# Properties

 Degree $12$ Conductor $2.754\times 10^{21}$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·13-s + 27-s − 3·31-s − 12·107-s + 3·113-s − 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
 L(s)  = 1 − 3·13-s + 27-s − 3·31-s − 12·107-s + 3·113-s − 3·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$12$$ Conductor: $$2^{30} \cdot 3^{12} \cdot 13^{6}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{3744} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(12,\ 2^{30} \cdot 3^{12} \cdot 13^{6} ,\ ( \ : [0]^{6} ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.03498185796$$ $$L(\frac12)$$ $$\approx$$ $$0.03498185796$$ $$L(1)$$ 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^{3} + T^{6}$$
13 $$( 1 + T + T^{2} )^{3}$$
good5 $$( 1 + T^{3} + T^{6} )^{2}$$
7 $$( 1 - T^{3} + T^{6} )^{2}$$
11 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
17 $$( 1 + T^{3} + T^{6} )^{2}$$
19 $$( 1 - T )^{6}( 1 + T )^{6}$$
23 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
29 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
31 $$( 1 + T )^{6}( 1 - T + T^{2} )^{3}$$
37 $$( 1 + T^{3} + T^{6} )^{2}$$
41 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
43 $$( 1 - T^{3} + T^{6} )^{2}$$
47 $$( 1 - T^{3} + T^{6} )^{2}$$
53 $$( 1 - T )^{6}( 1 + T )^{6}$$
59 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
61 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
67 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
71 $$( 1 - T^{3} + T^{6} )^{2}$$
73 $$( 1 - T )^{6}( 1 + T )^{6}$$
79 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
83 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
89 $$( 1 - T )^{6}( 1 + T )^{6}$$
97 $$( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$