# Properties

 Degree $8$ Conductor $252047376$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 4·3-s + 4-s − 2·5-s − 8·6-s + 2·7-s + 2·8-s + 6·9-s + 4·10-s − 4·11-s + 4·12-s − 4·14-s − 8·15-s − 4·16-s + 8·17-s − 12·18-s + 20·19-s − 2·20-s + 8·21-s + 8·22-s + 2·23-s + 8·24-s + 5·25-s − 4·27-s + 2·28-s + 4·29-s + 16·30-s + ⋯
 L(s)  = 1 − 1.41·2-s + 2.30·3-s + 1/2·4-s − 0.894·5-s − 3.26·6-s + 0.755·7-s + 0.707·8-s + 2·9-s + 1.26·10-s − 1.20·11-s + 1.15·12-s − 1.06·14-s − 2.06·15-s − 16-s + 1.94·17-s − 2.82·18-s + 4.58·19-s − 0.447·20-s + 1.74·21-s + 1.70·22-s + 0.417·23-s + 1.63·24-s + 25-s − 0.769·27-s + 0.377·28-s + 0.742·29-s + 2.92·30-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{126} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.04078$$ $$L(\frac12)$$ $$\approx$$ $$1.04078$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T + T^{2} )^{2}$$
3$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 - T + T^{2} )^{2}$$
good5$D_4\times C_2$ $$1 + 2 T - T^{2} - 2 p T^{3} - 4 p T^{4} - 2 p^{2} T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^3$ $$1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
19$D_{4}$ $$( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 80 p T^{5} - 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2^2$ $$( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_{4}$ $$( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^3$ $$1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2^3$ $$1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8}$$
53$D_{4}$ $$( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 + 18 T + 127 T^{2} + 1350 T^{3} + 15324 T^{4} + 1350 p T^{5} + 127 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 640 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_{4}$ $$( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 90 p T^{5} - 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2^2$ $$( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_{4}$ $$( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 80 p T^{5} - 158 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$