# Properties

 Degree $8$ Conductor $1.626\times 10^{13}$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s − 2·3-s + 10·4-s − 8·6-s + 20·8-s + 9-s + 3·11-s − 20·12-s + 35·16-s − 2·17-s + 4·18-s − 2·19-s + 12·22-s − 40·24-s + 4·25-s + 56·32-s − 6·33-s − 8·34-s + 10·36-s − 8·38-s − 2·41-s − 2·43-s + 30·44-s − 70·48-s − 49-s + 16·50-s + 4·51-s + ⋯
 L(s)  = 1 + 4·2-s − 2·3-s + 10·4-s − 8·6-s + 20·8-s + 9-s + 3·11-s − 20·12-s + 35·16-s − 2·17-s + 4·18-s − 2·19-s + 12·22-s − 40·24-s + 4·25-s + 56·32-s − 6·33-s − 8·34-s + 10·36-s − 8·38-s − 2·41-s − 2·43-s + 30·44-s − 70·48-s − 49-s + 16·50-s + 4·51-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 251^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 251^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 251^{4}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{2008} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 251^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$13.37437553$$ $$L(\frac12)$$ $$\approx$$ $$13.37437553$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 - T )^{4}$$
251$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
good3$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
5$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
7$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
11$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
13$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
17$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
19$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
23$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
29$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
31$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
37$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
41$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
43$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
53$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
59$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
61$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
67$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
71$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
73$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
79$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
83$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
89$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
97$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$