# Properties

 Label 8-637e4-1.1-c1e4-0-4 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 − 4·3-s + 4-s + 4·9-s + 12·11-s − 4·12-s − 4·13-s + 4·16-s − 12·17-s − 8·19-s − 6·23-s + 7·25-s + 4·27-s + 6·29-s − 2·31-s − 48·33-s + 4·36-s + 14·37-s + 16·39-s − 10·43-s + 12·44-s + 12·47-s − 16·48-s + 48·51-s − 4·52-s + 6·53-s + 32·57-s + 18·59-s + ⋯
 L(s)  = 1 − 2.30·3-s + 1/2·4-s + 4/3·9-s + 3.61·11-s − 1.15·12-s − 1.10·13-s + 16-s − 2.91·17-s − 1.83·19-s − 1.25·23-s + 7/5·25-s + 0.769·27-s + 1.11·29-s − 0.359·31-s − 8.35·33-s + 2/3·36-s + 2.30·37-s + 2.56·39-s − 1.52·43-s + 1.80·44-s + 1.75·47-s − 2.30·48-s + 6.72·51-s − 0.554·52-s + 0.824·53-s + 4.23·57-s + 2.34·59-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

 $$L(1)$$ $$\approx$$ $$0.5753926190$$ $$L(\frac12)$$ $$\approx$$ $$0.5753926190$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
good2$C_2^3$ $$1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8}$$
3$D_{4}$ $$( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
5$C_2^3$ $$1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 12 T + 77 T^{2} + 396 T^{3} + 1752 T^{4} + 396 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
23$D_4\times C_2$ $$1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 2 T - 32 T^{2} - 52 T^{3} + 211 T^{4} - 52 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^2$ $$( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^3$ $$1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 12 T + 62 T^{2} + 144 T^{3} - 2253 T^{4} + 144 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 6 T - 31 T^{2} + 234 T^{3} - 228 T^{4} + 234 p T^{5} - 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 18 T + 128 T^{2} - 1404 T^{3} + 15819 T^{4} - 1404 p T^{5} + 128 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
61$D_{4}$ $$( 1 - 20 T + 195 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_{4}$ $$( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 4 T - 107 T^{2} + 92 T^{3} + 8632 T^{4} + 92 p T^{5} - 107 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 22 T + 232 T^{2} + 2068 T^{3} + 19027 T^{4} + 2068 p T^{5} + 232 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 12 T - 22 T^{2} + 144 T^{3} + 6819 T^{4} + 144 p T^{5} - 22 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 + 8 T - 38 T^{2} - 736 T^{3} - 5213 T^{4} - 736 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$