# Properties

 Label 8-285e4-1.1-c1e4-0-0 Degree $8$ Conductor $6597500625$ Sign $1$ Analytic cond. $26.8217$ Root an. cond. $1.50855$ 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 + 3·4-s + 2·5-s − 4·6-s − 4·7-s − 2·8-s + 9-s − 4·10-s + 6·12-s − 2·13-s + 8·14-s + 4·15-s + 8·17-s − 2·18-s + 16·19-s + 6·20-s − 8·21-s − 4·23-s − 4·24-s + 25-s + 4·26-s − 2·27-s − 12·28-s − 8·29-s − 8·30-s − 20·31-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.51·7-s − 0.707·8-s + 1/3·9-s − 1.26·10-s + 1.73·12-s − 0.554·13-s + 2.13·14-s + 1.03·15-s + 1.94·17-s − 0.471·18-s + 3.67·19-s + 1.34·20-s − 1.74·21-s − 0.834·23-s − 0.816·24-s + 1/5·25-s + 0.784·26-s − 0.384·27-s − 2.26·28-s − 1.48·29-s − 1.46·30-s − 3.59·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(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(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: $$3^{4} \cdot 5^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$26.8217$$ Root analytic conductor: $$1.50855$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.301225439$$ $$L(\frac12)$$ $$\approx$$ $$1.301225439$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - T + T^{2} )^{2}$$
5$C_2$ $$( 1 - T + T^{2} )^{2}$$
19$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
good2$D_4\times C_2$ $$1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8}$$
7$D_{4}$ $$( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 14 T^{2} + p^{2} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 + 2 T - 15 T^{2} - 14 T^{3} + 140 T^{4} - 14 p T^{5} - 15 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 - 8 T + 22 T^{2} - 64 T^{3} + 387 T^{4} - 64 p T^{5} + 22 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
23$C_4\times C_2$ $$1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 + 8 T + 22 T^{2} - 128 T^{3} - 933 T^{4} - 128 p T^{5} + 22 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2$ $$( 1 + 5 T + p T^{2} )^{4}$$
37$D_{4}$ $$( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^3$ $$1 - 74 T^{2} + 3795 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 10 T - 3 T^{2} + 170 T^{3} + 4460 T^{4} + 170 p T^{5} - 3 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 12 T + 22 T^{2} - 336 T^{3} + 5907 T^{4} - 336 p T^{5} + 22 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 + 6 T + 33 T^{2} - 714 T^{3} - 5908 T^{4} - 714 p T^{5} + 33 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 6 T - 35 T^{2} + 378 T^{3} - 1860 T^{4} + 378 p T^{5} - 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 2 T - 71 T^{2} - 142 T^{3} + 4 T^{4} - 142 p T^{5} - 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 18 T + 117 T^{2} - 882 T^{3} + 11012 T^{4} - 882 p T^{5} + 117 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
89$D_4\times C_2$ $$1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 448 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$