# Properties

 Label 8-1805e4-1.1-c1e4-0-2 Degree $8$ Conductor $1.061\times 10^{13}$ Sign $1$ Analytic cond. $43153.6$ Root an. cond. $3.79644$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s + 4·5-s + 8·7-s + 8·11-s − 3·16-s + 8·17-s − 8·20-s + 8·23-s + 10·25-s − 16·28-s + 32·35-s − 8·43-s − 16·44-s + 8·47-s + 28·49-s + 32·55-s + 12·64-s − 16·68-s − 24·73-s + 64·77-s − 12·80-s − 10·81-s + 24·83-s + 32·85-s − 16·92-s − 20·100-s − 32·101-s + ⋯
 L(s)  = 1 − 4-s + 1.78·5-s + 3.02·7-s + 2.41·11-s − 3/4·16-s + 1.94·17-s − 1.78·20-s + 1.66·23-s + 2·25-s − 3.02·28-s + 5.40·35-s − 1.21·43-s − 2.41·44-s + 1.16·47-s + 4·49-s + 4.31·55-s + 3/2·64-s − 1.94·68-s − 2.80·73-s + 7.29·77-s − 1.34·80-s − 1.11·81-s + 2.63·83-s + 3.47·85-s − 1.66·92-s − 2·100-s − 3.18·101-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5^{4} \cdot 19^{8}$$ Sign: $1$ Analytic conductor: $$43153.6$$ Root analytic conductor: $$3.79644$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$12.81782302$$ $$L(\frac12)$$ $$\approx$$ $$12.81782302$$ $$L(\frac{3}{2})$$ 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)$
bad5$C_1$ $$( 1 - T )^{4}$$
19 $$1$$
good2$C_2^2 \wr C_2$ $$1 + p T^{2} + 7 T^{4} + p^{3} T^{6} + p^{4} T^{8}$$
3$D_4\times C_2$ $$1 + 10 T^{4} + p^{4} T^{8}$$
7$C_4$ $$( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
13$C_2^2 \wr C_2$ $$1 + 32 T^{2} + 522 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8}$$
17$C_4$ $$( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_4$ $$( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 + 36 T^{2} + 854 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8}$$
31$C_2^2 \wr C_2$ $$1 + 76 T^{2} + 3238 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2^2 \wr C_2$ $$1 + 96 T^{2} + 4394 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2$ $$( 1 + p T^{2} )^{4}$$
43$D_{4}$ $$( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
47$D_{4}$ $$( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
53$C_2^2 \wr C_2$ $$1 + 128 T^{2} + 9322 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2 \wr C_2$ $$1 + 44 T^{2} + 5398 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 + 114 T^{2} + p^{2} T^{4} )^{2}$$
67$C_2^2 \wr C_2$ $$1 + 192 T^{2} + 18122 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2^2 \wr C_2$ $$1 - 20 T^{2} + 9030 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
73$D_{4}$ $$( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^2 \wr C_2$ $$1 + 124 T^{2} + 14278 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
89$C_2^2 \wr C_2$ $$1 + 20 T^{2} + 9670 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2 \wr C_2$ $$1 + 368 T^{2} + 52602 T^{4} + 368 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$