# Properties

 Label 8-799e4-1.1-c0e4-0-2 Degree $8$ Conductor $407555836801$ Sign $1$ Analytic cond. $0.0252822$ Root an. cond. $0.631468$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s + 10·9-s − 2·16-s − 4·17-s + 20·27-s − 8·48-s − 2·49-s − 16·51-s − 4·53-s − 4·61-s + 4·71-s + 4·79-s + 34·81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s − 20·144-s − 8·147-s + 149-s + 151-s − 40·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯
 L(s)  = 1 + 4·3-s + 10·9-s − 2·16-s − 4·17-s + 20·27-s − 8·48-s − 2·49-s − 16·51-s − 4·53-s − 4·61-s + 4·71-s + 4·79-s + 34·81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s − 20·144-s − 8·147-s + 149-s + 151-s − 40·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$17^{4} \cdot 47^{4}$$ Sign: $1$ Analytic conductor: $$0.0252822$$ Root analytic conductor: $$0.631468$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{799} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 17^{4} \cdot 47^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.986613724$$ $$L(\frac12)$$ $$\approx$$ $$2.986613724$$ $$L(1)$$ 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)$
bad17$C_1$ $$( 1 + T )^{4}$$
47$C_2$ $$( 1 + T^{2} )^{2}$$
good2$C_2^2$ $$( 1 + T^{4} )^{2}$$
3$C_1$$\times$$C_2^2$ $$( 1 - T )^{4}( 1 + T^{4} )$$
5$C_4\times C_2$ $$1 + T^{8}$$
7$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
11$C_4\times C_2$ $$1 + T^{8}$$
13$C_2$ $$( 1 + T^{2} )^{4}$$
19$C_2^2$ $$( 1 + T^{4} )^{2}$$
23$C_4\times C_2$ $$1 + T^{8}$$
29$C_4\times C_2$ $$1 + T^{8}$$
31$C_4\times C_2$ $$1 + T^{8}$$
37$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
41$C_4\times C_2$ $$1 + T^{8}$$
43$C_2^2$ $$( 1 + T^{4} )^{2}$$
53$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 + T^{2} )^{2}$$
59$C_2^2$ $$( 1 + T^{4} )^{2}$$
61$C_1$$\times$$C_2^2$ $$( 1 + T )^{4}( 1 + T^{4} )$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
71$C_1$$\times$$C_2^2$ $$( 1 - T )^{4}( 1 + T^{4} )$$
73$C_4\times C_2$ $$1 + T^{8}$$
79$C_1$$\times$$C_2^2$ $$( 1 - T )^{4}( 1 + T^{4} )$$
83$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 + T^{2} )^{2}$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
97$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 + T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$