# Properties

 Label 8-570e4-1.1-c1e4-0-3 Degree $8$ Conductor $105560010000$ Sign $1$ Analytic cond. $429.148$ Root an. cond. $2.13341$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·3-s + 4-s + 2·5-s + 4·6-s − 2·8-s + 9-s + 4·10-s − 4·11-s + 2·12-s + 4·15-s − 4·16-s + 6·17-s + 2·18-s − 12·19-s + 2·20-s − 8·22-s − 2·23-s − 4·24-s + 25-s − 2·27-s + 2·29-s + 8·30-s + 8·31-s − 2·32-s − 8·33-s + 12·34-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 0.707·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s + 0.577·12-s + 1.03·15-s − 16-s + 1.45·17-s + 0.471·18-s − 2.75·19-s + 0.447·20-s − 1.70·22-s − 0.417·23-s − 0.816·24-s + 1/5·25-s − 0.384·27-s + 0.371·29-s + 1.46·30-s + 1.43·31-s − 0.353·32-s − 1.39·33-s + 2.05·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{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^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$429.148$$ Root analytic conductor: $$2.13341$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{570} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$6.422125789$$ $$L(\frac12)$$ $$\approx$$ $$6.422125789$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - T + T^{2} )^{2}$$
5$C_2$ $$( 1 - T + T^{2} )^{2}$$
19$C_2^2$ $$1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
good7$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
11$D_{4}$ $$( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 6 T - 12 T^{3} + 395 T^{4} - 12 p T^{5} - 6 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 - 2 T - 48 T^{2} + 12 T^{3} + 1747 T^{4} + 12 p T^{5} - 48 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
31$D_{4}$ $$( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_{4}$ $$( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 12 T + 33 T^{2} - 348 T^{3} + 4736 T^{4} - 348 p T^{5} + 33 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2^2$ $$( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 4 T + 30 T^{2} - 432 T^{3} - 2765 T^{4} - 432 p T^{5} + 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 4 T - 87 T^{2} + 12 T^{3} + 6952 T^{4} + 12 p T^{5} - 87 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2^3$ $$1 - 90 T^{2} + 4619 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 2 T - 56 T^{2} + 124 T^{3} - 365 T^{4} + 124 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 14 T + 20 T^{2} - 588 T^{3} + 13355 T^{4} - 588 p T^{5} + 20 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 22 T + 228 T^{2} - 2508 T^{3} + 26131 T^{4} - 2508 p T^{5} + 228 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 + 22 T + 224 T^{2} + 2508 T^{3} + 26939 T^{4} + 2508 p T^{5} + 224 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2^3$ $$1 - 130 T^{2} + 10659 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
89$C_2^3$ $$1 - 3 T^{2} - 7912 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 + 2 T - 184 T^{2} - 12 T^{3} + 25547 T^{4} - 12 p T^{5} - 184 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$