# Properties

 Label 8-50e8-1.1-c0e4-0-3 Degree $8$ Conductor $3.906\times 10^{13}$ Sign $1$ Analytic cond. $2.42319$ Root an. cond. $1.11698$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s − 9-s − 3·13-s − 3·17-s − 18-s − 3·26-s + 3·29-s − 32-s − 3·34-s + 2·37-s + 3·41-s + 4·49-s + 2·53-s + 3·58-s + 3·61-s − 64-s − 3·73-s + 2·74-s + 3·82-s − 2·89-s − 3·97-s + 4·98-s − 2·101-s + 2·106-s + 3·109-s + 2·113-s + 3·117-s + ⋯
 L(s)  = 1 + 2-s − 9-s − 3·13-s − 3·17-s − 18-s − 3·26-s + 3·29-s − 32-s − 3·34-s + 2·37-s + 3·41-s + 4·49-s + 2·53-s + 3·58-s + 3·61-s − 64-s − 3·73-s + 2·74-s + 3·82-s − 2·89-s − 3·97-s + 4·98-s − 2·101-s + 2·106-s + 3·109-s + 2·113-s + 3·117-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\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 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$2.42319$$ Root analytic conductor: $$1.11698$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2500} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.529525769$$ $$L(\frac12)$$ $$\approx$$ $$1.529525769$$ $$L(1)$$ 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$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
5 $$1$$
good3$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
7$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
11$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
13$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
17$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
19$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
23$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
29$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
31$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
37$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
41$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
47$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
53$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
59$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
61$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
67$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
71$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
73$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
79$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
83$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
89$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
97$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$