# Properties

 Label 8-1805e4-1.1-c0e4-0-1 Degree $8$ Conductor $1.061\times 10^{13}$ Sign $1$ Analytic cond. $0.658472$ Root an. cond. $0.949111$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·5-s + 4·7-s + 16-s − 2·17-s + 2·23-s + 25-s + 8·35-s − 2·43-s − 2·47-s + 8·49-s − 2·73-s + 2·80-s + 81-s + 4·83-s − 4·85-s + 4·112-s + 4·115-s − 8·119-s − 4·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 + 2·5-s + 4·7-s + 16-s − 2·17-s + 2·23-s + 25-s + 8·35-s − 2·43-s − 2·47-s + 8·49-s − 2·73-s + 2·80-s + 81-s + 4·83-s − 4·85-s + 4·112-s + 4·115-s − 8·119-s − 4·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5^{4} \cdot 19^{8}$$ Sign: $1$ Analytic conductor: $$0.658472$$ Root analytic conductor: $$0.949111$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$3.575505740$$ $$L(\frac12)$$ $$\approx$$ $$3.575505740$$ $$L(1)$$ 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)$
bad5$C_2$ $$( 1 - T + T^{2} )^{2}$$
19 $$1$$
good2$C_2^3$ $$1 - T^{4} + T^{8}$$
3$C_2^3$ $$1 - T^{4} + T^{8}$$
7$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T^{2} )^{2}$$
11$C_2$ $$( 1 + T^{2} )^{4}$$
13$C_2^3$ $$1 - T^{4} + T^{8}$$
17$C_2$$\times$$C_2^2$ $$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
23$C_2$$\times$$C_2^2$ $$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
29$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
31$C_2$ $$( 1 + T^{2} )^{4}$$
37$C_2^2$ $$( 1 + T^{4} )^{2}$$
41$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
43$C_2$$\times$$C_2^2$ $$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
47$C_2$$\times$$C_2^2$ $$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
53$C_2^3$ $$1 - T^{4} + T^{8}$$
59$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
61$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
67$C_2^3$ $$1 - T^{4} + T^{8}$$
71$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
73$C_2$$\times$$C_2^2$ $$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
79$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
83$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T^{2} )^{2}$$
89$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
97$C_2^3$ $$1 - T^{4} + T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$