# Properties

 Label 8-637e4-1.1-c1e4-0-24 Degree $8$ Conductor $164648481361$ Sign $1$ Analytic cond. $669.369$ Root an. cond. $2.25532$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 3·2-s + 5·4-s + 6·8-s + 9-s + 6·11-s + 4·13-s + 4·16-s − 6·17-s + 3·18-s + 6·19-s + 18·22-s + 12·23-s + 5·25-s + 12·26-s + 10·31-s − 18·34-s + 5·36-s + 4·37-s + 18·38-s − 32·43-s + 30·44-s + 36·46-s − 6·47-s + 15·50-s + 20·52-s + 6·53-s + 6·59-s + ⋯
 L(s)  = 1 + 2.12·2-s + 5/2·4-s + 2.12·8-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 16-s − 1.45·17-s + 0.707·18-s + 1.37·19-s + 3.83·22-s + 2.50·23-s + 25-s + 2.35·26-s + 1.79·31-s − 3.08·34-s + 5/6·36-s + 0.657·37-s + 2.91·38-s − 4.87·43-s + 4.52·44-s + 5.30·46-s − 0.875·47-s + 2.12·50-s + 2.77·52-s + 0.824·53-s + 0.781·59-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$7^{8} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$669.369$$ Root analytic conductor: $$2.25532$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{637} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$15.69775001$$ $$L(\frac12)$$ $$\approx$$ $$15.69775001$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 - T )^{4}$$
good2$C_2^2$$\times$$C_2^2$ $$( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 - T^{2} + p^{2} T^{4} )$$
3$C_2^3$ $$1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8}$$
5$C_2$$\times$$C_2^2$ $$( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} )$$
11$C_2^2$ $$( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 6 T + 13 T^{2} - 66 T^{3} - 372 T^{4} - 66 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + 38 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 4 T - 17 T^{2} + 164 T^{3} - 872 T^{4} + 164 p T^{5} - 17 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2$ $$( 1 + 62 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
47$D_4\times C_2$ $$1 + 6 T - p T^{2} - 66 T^{3} + 2988 T^{4} - 66 p T^{5} - p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 6 T - 71 T^{2} + 66 T^{3} + 5844 T^{4} + 66 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 62 T^{2} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 8 T - 53 T^{2} + 232 T^{3} + 3688 T^{4} + 232 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 8 T - 65 T^{2} - 232 T^{3} + 5344 T^{4} - 232 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2$ $$( 1 + p T^{2} )^{4}$$
89$C_2^3$ $$1 - 173 T^{2} + 22008 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$