# Properties

 Label 8-3200e4-1.1-c0e4-0-6 Degree $8$ Conductor $1.049\times 10^{14}$ Sign $1$ Analytic cond. $6.50471$ Root an. cond. $1.26372$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5-s + 9-s + 2·17-s − 5·37-s − 2·41-s + 45-s + 4·49-s + 5·53-s + 2·73-s + 2·85-s − 3·89-s + 2·97-s + 3·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 5-s + 9-s + 2·17-s − 5·37-s − 2·41-s + 45-s + 4·49-s + 5·53-s + 2·73-s + 2·85-s − 3·89-s + 2·97-s + 3·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$6.50471$$ Root analytic conductor: $$1.26372$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.708673522$$ $$L(\frac12)$$ $$\approx$$ $$2.708673522$$ $$L(1)$$ 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 $$1$$
5$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
good3$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
7$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
11$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
13$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
17$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
19$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
23$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
29$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
31$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
37$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
41$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
43$C_2$ $$( 1 + T^{2} )^{4}$$
47$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
53$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
59$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
61$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
67$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
71$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
73$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
79$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
83$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
89$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
97$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$