# Properties

 Label 8-124e4-1.1-c0e4-0-0 Degree $8$ Conductor $236421376$ Sign $1$ Analytic cond. $1.46661\times 10^{-5}$ Root an. cond. $0.248765$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 2·5-s − 9-s + 2·13-s + 3·16-s − 2·17-s + 4·20-s + 3·25-s + 2·36-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 4·52-s + 2·53-s − 4·64-s − 4·65-s + 4·68-s − 2·73-s − 6·80-s + 81-s + 4·85-s − 6·100-s + 2·113-s − 2·117-s − 121-s − 6·125-s + ⋯
 L(s)  = 1 − 2·4-s − 2·5-s − 9-s + 2·13-s + 3·16-s − 2·17-s + 4·20-s + 3·25-s + 2·36-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 4·52-s + 2·53-s − 4·64-s − 4·65-s + 4·68-s − 2·73-s − 6·80-s + 81-s + 4·85-s − 6·100-s + 2·113-s − 2·117-s − 121-s − 6·125-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 31^{4}$$ Sign: $1$ Analytic conductor: $$1.46661\times 10^{-5}$$ Root analytic conductor: $$0.248765$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{124} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 31^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.07615071625$$ $$L(\frac12)$$ $$\approx$$ $$0.07615071625$$ $$L(1)$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
31$C_2$ $$( 1 + T^{2} )^{2}$$
good3$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
5$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
7$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
11$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
13$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
17$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
19$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
23$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
29$C_2$ $$( 1 + T^{2} )^{4}$$
37$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
41$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
43$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
47$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
53$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
59$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
61$C_2$ $$( 1 + T^{2} )^{4}$$
67$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
71$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
73$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
79$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
83$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
89$C_2$ $$( 1 + T^{2} )^{4}$$
97$C_2$ $$( 1 + T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$