# Properties

 Label 8-5586e4-1.1-c1e4-0-0 Degree $8$ Conductor $9.737\times 10^{14}$ Sign $1$ Analytic cond. $3.95833\times 10^{6}$ Root an. cond. $6.67865$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s − 4·3-s + 10·4-s + 16·6-s − 20·8-s + 10·9-s − 2·11-s − 40·12-s + 35·16-s + 10·17-s − 40·18-s + 4·19-s + 8·22-s − 5·23-s + 80·24-s − 8·25-s − 20·27-s − 3·29-s + 9·31-s − 56·32-s + 8·33-s − 40·34-s + 100·36-s − 14·37-s − 16·38-s + 4·41-s + 21·43-s + ⋯
 L(s)  = 1 − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s − 7.07·8-s + 10/3·9-s − 0.603·11-s − 11.5·12-s + 35/4·16-s + 2.42·17-s − 9.42·18-s + 0.917·19-s + 1.70·22-s − 1.04·23-s + 16.3·24-s − 8/5·25-s − 3.84·27-s − 0.557·29-s + 1.61·31-s − 9.89·32-s + 1.39·33-s − 6.85·34-s + 50/3·36-s − 2.30·37-s − 2.59·38-s + 0.624·41-s + 3.20·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \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 7^{8} \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: $$2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$3.95833\times 10^{6}$$ Root analytic conductor: $$6.67865$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{5586} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8458527037$$ $$L(\frac12)$$ $$\approx$$ $$0.8458527037$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 + T )^{4}$$
3$C_1$ $$( 1 + T )^{4}$$
7 $$1$$
19$C_1$ $$( 1 - T )^{4}$$
good5$S_4\times C_2$ $$1 + 8 T^{2} + 6 T^{3} + 51 T^{4} + 6 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 + 2 T + 32 T^{2} + 4 p T^{3} + 469 T^{4} + 4 p^{2} T^{5} + 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 + 4 T^{2} - 48 T^{3} + 102 T^{4} - 48 p T^{5} + 4 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 - 10 T + 62 T^{2} - 256 T^{3} + 1078 T^{4} - 256 p T^{5} + 62 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 + 5 T + 32 T^{2} + 155 T^{3} + 1336 T^{4} + 155 p T^{5} + 32 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 + 3 T + 35 T^{2} + 126 T^{3} + 1644 T^{4} + 126 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 - 9 T + 91 T^{2} - 360 T^{3} + 2652 T^{4} - 360 p T^{5} + 91 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 + 14 T + 190 T^{2} + 1532 T^{3} + 11314 T^{4} + 1532 p T^{5} + 190 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 - 4 T + 68 T^{2} + 14 T^{3} + 1870 T^{4} + 14 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 - 21 T + 286 T^{2} - 2685 T^{3} + 20244 T^{4} - 2685 p T^{5} + 286 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 - 7 T + 164 T^{2} - 859 T^{3} + 11254 T^{4} - 859 p T^{5} + 164 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 + 7 T + 119 T^{2} + 694 T^{3} + 7972 T^{4} + 694 p T^{5} + 119 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 - 7 T + 53 T^{2} - 586 T^{3} + 8494 T^{4} - 586 p T^{5} + 53 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 - 23 T + 430 T^{2} - 4829 T^{3} + 45730 T^{4} - 4829 p T^{5} + 430 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 - 6 T + 28 T^{2} - 198 T^{3} + 2742 T^{4} - 198 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - 2 T + 158 T^{2} - 380 T^{3} + 12382 T^{4} - 380 p T^{5} + 158 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 + 5 T + 106 T^{2} - 145 T^{3} + 3142 T^{4} - 145 p T^{5} + 106 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 + 11 T + 319 T^{2} + 2456 T^{3} + 38092 T^{4} + 2456 p T^{5} + 319 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 - 14 T + 272 T^{2} - 3200 T^{3} + 31417 T^{4} - 3200 p T^{5} + 272 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 10 T + 200 T^{2} - 8 p T^{3} + 14914 T^{4} - 8 p^{2} T^{5} + 200 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 + T + 319 T^{2} + 232 T^{3} + 44116 T^{4} + 232 p T^{5} + 319 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$