# Properties

 Label 8-799e4-1.1-c0e4-0-0 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 + 2·2-s + 4-s − 9-s − 17-s − 2·18-s − 4·25-s − 2·32-s − 2·34-s − 36-s + 4·47-s − 49-s − 8·50-s − 2·53-s + 2·59-s − 4·64-s − 68-s + 8·83-s − 2·89-s + 8·94-s − 2·98-s − 4·100-s + 2·101-s + 2·103-s − 4·106-s + 4·118-s − 4·121-s + 127-s + ⋯
 L(s)  = 1 + 2·2-s + 4-s − 9-s − 17-s − 2·18-s − 4·25-s − 2·32-s − 2·34-s − 36-s + 4·47-s − 49-s − 8·50-s − 2·53-s + 2·59-s − 4·64-s − 68-s + 8·83-s − 2·89-s + 8·94-s − 2·98-s − 4·100-s + 2·101-s + 2·103-s − 4·106-s + 4·118-s − 4·121-s + 127-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: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 17^{4} \cdot 47^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.308381307$$ $$L(\frac12)$$ $$\approx$$ $$1.308381307$$ $$L(1)$$ 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$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
47$C_1$ $$( 1 - T )^{4}$$
good2$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
3$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
5$C_2$ $$( 1 + T^{2} )^{4}$$
7$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
11$C_2$ $$( 1 + T^{2} )^{4}$$
13$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
23$C_2$ $$( 1 + T^{2} )^{4}$$
29$C_2$ $$( 1 + T^{2} )^{4}$$
31$C_2$ $$( 1 + T^{2} )^{4}$$
37$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
41$C_2$ $$( 1 + T^{2} )^{4}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
53$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
59$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
61$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
71$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
73$C_2$ $$( 1 + T^{2} )^{4}$$
79$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
83$C_1$ $$( 1 - T )^{8}$$
89$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
97$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$