# Properties

 Label 8-2e32-1.1-c1e4-0-0 Degree $8$ Conductor $4294967296$ Sign $1$ Analytic cond. $17.4609$ Root an. cond. $1.42974$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·5-s − 4·7-s − 2·9-s − 8·11-s − 4·13-s − 8·19-s − 12·23-s + 10·25-s + 8·27-s + 4·29-s + 16·31-s − 16·35-s − 4·37-s − 12·41-s + 16·43-s − 8·45-s + 8·49-s − 4·53-s − 32·55-s − 16·59-s − 4·61-s + 8·63-s − 16·65-s − 8·67-s + 12·71-s + 28·73-s + 32·77-s + ⋯
 L(s)  = 1 + 1.78·5-s − 1.51·7-s − 2/3·9-s − 2.41·11-s − 1.10·13-s − 1.83·19-s − 2.50·23-s + 2·25-s + 1.53·27-s + 0.742·29-s + 2.87·31-s − 2.70·35-s − 0.657·37-s − 1.87·41-s + 2.43·43-s − 1.19·45-s + 8/7·49-s − 0.549·53-s − 4.31·55-s − 2.08·59-s − 0.512·61-s + 1.00·63-s − 1.98·65-s − 0.977·67-s + 1.42·71-s + 3.27·73-s + 3.64·77-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{32}$$ Sign: $1$ Analytic conductor: $$17.4609$$ Root analytic conductor: $$1.42974$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{256} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{32} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7096370182$$ $$L(\frac12)$$ $$\approx$$ $$0.7096370182$$ $$L(\frac{3}{2})$$ 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 $$1$$
good3$C_2$$\times$$C_2^2$ $$( 1 + 2 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )$$
5$C_2$$\times$$C_2^2$ $$( 1 - 2 T + p T^{2} )^{2}( 1 - 8 T^{2} + p^{2} T^{4} )$$
7$C_2^2$ $$( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2$$\times$$C_2^2$ $$( 1 + 6 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )$$
13$D_4\times C_2$ $$1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2^2$ $$( 1 - 26 T^{2} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 + 8 T + 18 T^{2} - 160 T^{3} - 1246 T^{4} - 160 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 - 4 T + 6 T^{2} + 204 T^{3} - 830 T^{4} + 204 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
37$D_4\times C_2$ $$1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 516 T^{3} + 3694 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 16 T + 162 T^{2} - 1384 T^{3} + 10178 T^{4} - 1384 p T^{5} + 162 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 + 4 T + 54 T^{2} + 708 T^{3} + 3490 T^{4} + 708 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + 16 T + 114 T^{2} + 696 T^{3} + 4834 T^{4} + 696 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 + 8 T + 18 T^{2} - 736 T^{3} - 5854 T^{4} - 736 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 876 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 122 T^{2} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 16 T + 114 T^{2} - 792 T^{3} + 6370 T^{4} - 792 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 12 T + 72 T^{2} - 516 T^{3} + 1582 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$