# Properties

 Degree $8$ Conductor $37822859361$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3·4-s + 4·5-s − 2·8-s − 8·10-s − 4·11-s + 16·13-s + 4·17-s + 12·20-s + 8·22-s − 4·23-s + 12·25-s − 32·26-s + 16·29-s + 8·31-s + 6·32-s − 8·34-s + 8·37-s − 8·40-s − 8·41-s − 12·44-s + 8·46-s − 24·50-s + 48·52-s − 4·53-s − 16·55-s − 32·58-s + ⋯
 L(s)  = 1 − 1.41·2-s + 3/2·4-s + 1.78·5-s − 0.707·8-s − 2.52·10-s − 1.20·11-s + 4.43·13-s + 0.970·17-s + 2.68·20-s + 1.70·22-s − 0.834·23-s + 12/5·25-s − 6.27·26-s + 2.97·29-s + 1.43·31-s + 1.06·32-s − 1.37·34-s + 1.31·37-s − 1.26·40-s − 1.24·41-s − 1.80·44-s + 1.17·46-s − 3.39·50-s + 6.65·52-s − 0.549·53-s − 2.15·55-s − 4.20·58-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 7^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{441} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.94269$$ $$L(\frac12)$$ $$\approx$$ $$2.94269$$ $$L(\frac{3}{2})$$ 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)$
bad3 $$1$$
7 $$1$$
good2$D_4\times C_2$ $$1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}$$
5$D_4\times C_2$ $$1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
13$D_{4}$ $$( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 4 T - 4 T^{2} + 56 T^{3} - 161 T^{4} + 56 p T^{5} - 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2^3$ $$1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 4 T - 2 T^{2} - 112 T^{3} - 573 T^{4} - 112 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 64 p T^{5} - 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^2$ $$( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$D_{4}$ $$( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 + 54 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 64 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 736 p T^{5} + 88 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^3$ $$1 - 102 T^{2} + 5915 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 8 T - 656 T^{3} - 5905 T^{4} - 656 p T^{5} + 8 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 1640 p T^{5} + 140 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$