# Properties

 Label 8-3332e4-1.1-c0e4-0-9 Degree $8$ Conductor $1.233\times 10^{14}$ Sign $1$ Analytic cond. $7.64624$ Root an. cond. $1.28952$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 2·5-s + 8·13-s + 2·20-s + 2·25-s − 4·29-s − 2·37-s + 4·41-s + 8·52-s − 2·61-s − 64-s + 16·65-s + 2·73-s + 81-s + 4·89-s − 4·97-s + 2·100-s − 4·101-s − 2·109-s − 4·113-s − 4·116-s + 127-s + 131-s + 137-s + 139-s − 8·145-s − 2·148-s + ⋯
 L(s)  = 1 + 4-s + 2·5-s + 8·13-s + 2·20-s + 2·25-s − 4·29-s − 2·37-s + 4·41-s + 8·52-s − 2·61-s − 64-s + 16·65-s + 2·73-s + 81-s + 4·89-s − 4·97-s + 2·100-s − 4·101-s − 2·109-s − 4·113-s − 4·116-s + 127-s + 131-s + 137-s + 139-s − 8·145-s − 2·148-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\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 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 7^{8} \cdot 17^{4}$$ Sign: $1$ Analytic conductor: $$7.64624$$ Root analytic conductor: $$1.28952$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 7^{8} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$6.482892037$$ $$L(\frac12)$$ $$\approx$$ $$6.482892037$$ $$L(1)$$ 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$C_2^2$ $$1 - T^{2} + T^{4}$$
7 $$1$$
17$C_2^2$ $$1 - T^{2} + T^{4}$$
good3$C_2^3$ $$1 - T^{4} + T^{8}$$
5$C_2$$\times$$C_2^2$ $$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
11$C_2^3$ $$1 - T^{4} + T^{8}$$
13$C_1$ $$( 1 - T )^{8}$$
19$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
23$C_2^3$ $$1 - T^{4} + T^{8}$$
29$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 + T^{2} )^{2}$$
31$C_2^3$ $$1 - T^{4} + T^{8}$$
37$C_2$$\times$$C_2^2$ $$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
41$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T^{2} )^{2}$$
43$C_2$ $$( 1 + T^{2} )^{4}$$
47$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
53$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
59$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
61$C_2$$\times$$C_2^2$ $$( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
67$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
71$C_2^2$ $$( 1 + T^{4} )^{2}$$
73$C_2$$\times$$C_2^2$ $$( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
79$C_2^3$ $$1 - T^{4} + T^{8}$$
83$C_2$ $$( 1 + T^{2} )^{4}$$
89$C_2$ $$( 1 - T + T^{2} )^{4}$$
97$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 + T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$