# Properties

 Label 8-91e8-1.1-c1e4-0-7 Degree $8$ Conductor $4.703\times 10^{15}$ Sign $1$ Analytic cond. $1.91178\times 10^{7}$ Root an. cond. $8.13167$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 2·4-s − 20·8-s − 8·9-s + 4·11-s − 45·16-s − 32·18-s + 16·22-s − 12·23-s + 4·25-s − 16·29-s + 16·32-s − 16·36-s − 8·37-s + 12·43-s + 8·44-s − 48·46-s + 16·50-s − 12·53-s − 64·58-s + 204·64-s + 4·67-s + 24·71-s + 160·72-s − 32·74-s − 28·79-s + 30·81-s + ⋯
 L(s)  = 1 + 2.82·2-s + 4-s − 7.07·8-s − 8/3·9-s + 1.20·11-s − 11.2·16-s − 7.54·18-s + 3.41·22-s − 2.50·23-s + 4/5·25-s − 2.97·29-s + 2.82·32-s − 8/3·36-s − 1.31·37-s + 1.82·43-s + 1.20·44-s − 7.07·46-s + 2.26·50-s − 1.64·53-s − 8.40·58-s + 51/2·64-s + 0.488·67-s + 2.84·71-s + 18.8·72-s − 3.71·74-s − 3.15·79-s + 10/3·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$7^{8} \cdot 13^{8}$$ Sign: $1$ Analytic conductor: $$1.91178\times 10^{7}$$ Root analytic conductor: $$8.13167$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{8281} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 7^{8} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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)$
bad7 $$1$$
13 $$1$$
good2$C_2$ $$( 1 - T + p T^{2} )^{4}$$
3$C_2^2$ $$( 1 + 4 T^{2} + p^{2} T^{4} )^{2}$$
5$D_4\times C_2$ $$1 - 4 T^{2} + 31 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 36 T^{2} + 695 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 + 30 T^{2} + p^{2} T^{4} )^{2}$$
23$D_{4}$ $$( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
29$D_{4}$ $$( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 60 T^{2} + p^{2} T^{4} )^{2}$$
37$D_{4}$ $$( 1 + 4 T + 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 + 60 T^{2} + 2399 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 - 6 T + 72 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 86 T^{2} + p^{2} T^{4} )^{2}$$
53$D_{4}$ $$( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 + 80 T^{2} + 2674 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 172 T^{2} + 13711 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 - 2 T + 112 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
73$D_4\times C_2$ $$1 + 100 T^{2} + 127 p T^{4} + 100 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 68 T^{2} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 16 T^{2} + 14434 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 + 176 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$