# Properties

 Degree $8$ Conductor $2.560\times 10^{14}$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·9-s + 2·11-s − 2·19-s − 2·41-s + 49-s − 2·59-s + 10·81-s + 2·89-s − 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 8·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 − 4·9-s + 2·11-s − 2·19-s − 2·41-s + 49-s − 2·59-s + 10·81-s + 2·89-s − 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 8·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{20} \cdot 5^{12}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{4000} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{20} \cdot 5^{12} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1210823462$$ $$L(\frac12)$$ $$\approx$$ $$0.1210823462$$ $$L(1)$$ 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 $$1$$
5 $$1$$
good3$C_2$ $$( 1 + T^{2} )^{4}$$
7$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
11$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
13$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
17$C_2$ $$( 1 + T^{2} )^{4}$$
19$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
23$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
29$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
37$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
41$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
43$C_2$ $$( 1 + T^{2} )^{4}$$
47$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
53$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
59$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
61$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
67$C_2$ $$( 1 + T^{2} )^{4}$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
73$C_2$ $$( 1 + T^{2} )^{4}$$
79$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
83$C_2$ $$( 1 + T^{2} )^{4}$$
89$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
97$C_2$ $$( 1 + T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$