# Properties

 Label 8-34e8-1.1-c0e4-0-0 Degree $8$ Conductor $1.786\times 10^{12}$ Sign $1$ Analytic cond. $0.110779$ Root an. cond. $0.759551$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 16-s + 8·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
 L(s)  = 1 − 16-s + 8·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 17^{8}\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 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 17^{8}$$ Sign: $1$ Analytic conductor: $$0.110779$$ Root analytic conductor: $$0.759551$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1156} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 17^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8436205050$$ $$L(\frac12)$$ $$\approx$$ $$0.8436205050$$ $$L(1)$$ 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$C_2^2$ $$1 + T^{4}$$
17 $$1$$
good3$C_4\times C_2$ $$1 + T^{8}$$
5$C_4\times C_2$ $$1 + T^{8}$$
7$C_4\times C_2$ $$1 + T^{8}$$
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$C_4\times C_2$ $$1 + T^{8}$$
41$C_4\times C_2$ $$1 + T^{8}$$
43$C_2^2$ $$( 1 + T^{4} )^{2}$$
47$C_2$ $$( 1 + T^{2} )^{4}$$
53$C_2^2$ $$( 1 + T^{4} )^{2}$$
59$C_2^2$ $$( 1 + T^{4} )^{2}$$
61$C_4\times C_2$ $$1 + T^{8}$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
71$C_4\times C_2$ $$1 + T^{8}$$
73$C_4\times C_2$ $$1 + T^{8}$$
79$C_4\times C_2$ $$1 + T^{8}$$
83$C_2^2$ $$( 1 + T^{4} )^{2}$$
89$C_2$ $$( 1 + T^{2} )^{4}$$
97$C_4\times C_2$ $$1 + T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$