# Properties

 Label 8-475e4-1.1-c1e4-0-1 Degree $8$ Conductor $50906640625$ Sign $1$ Analytic cond. $206.958$ Root an. cond. $1.94753$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·7-s − 8·16-s + 14·17-s − 8·23-s + 2·43-s − 26·47-s + 18·49-s + 22·73-s − 18·81-s + 32·83-s + 48·112-s − 84·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 48·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 2.26·7-s − 2·16-s + 3.39·17-s − 1.66·23-s + 0.304·43-s − 3.79·47-s + 18/7·49-s + 2.57·73-s − 2·81-s + 3.51·83-s + 4.53·112-s − 7.70·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 3.78·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$5^{8} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$206.958$$ Root analytic conductor: $$1.94753$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{475} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6492179306$$ $$L(\frac12)$$ $$\approx$$ $$0.6492179306$$ $$L(\frac{3}{2})$$ 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 $$1$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
good2$C_2$ $$( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2}$$
3$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
7$C_2$$\times$$C_2^2$ $$( 1 + 3 T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} )$$
11$C_2^2$ $$( 1 + 3 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
17$C_2$$\times$$C_2^2$ $$( 1 - 7 T + p T^{2} )^{2}( 1 + 15 T^{2} + p^{2} T^{4} )$$
23$C_2$$\times$$C_2^2$ $$( 1 + 4 T + p T^{2} )^{2}( 1 - 30 T^{2} + p^{2} T^{4} )$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2$ $$( 1 - p T^{2} )^{4}$$
37$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
41$C_2$ $$( 1 - p T^{2} )^{4}$$
43$C_2$$\times$$C_2^2$ $$( 1 - T + p T^{2} )^{2}( 1 - 85 T^{2} + p^{2} T^{4} )$$
47$C_2$$\times$$C_2^2$ $$( 1 + 13 T + p T^{2} )^{2}( 1 + 75 T^{2} + p^{2} T^{4} )$$
53$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
59$C_2$ $$( 1 + p T^{2} )^{4}$$
61$C_2^2$ $$( 1 + 103 T^{2} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 - p T^{2} )^{4}$$
73$C_2$$\times$$C_2^2$ $$( 1 - 11 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} )$$
79$C_2$ $$( 1 + p T^{2} )^{4}$$
83$C_2$$\times$$C_2^2$ $$( 1 - 16 T + p T^{2} )^{2}( 1 + 90 T^{2} + p^{2} T^{4} )$$
89$C_2$ $$( 1 + p T^{2} )^{4}$$
97$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$