# Properties

 Label 8-637e4-1.1-c1e4-0-27 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-s + 3·4-s − 3·5-s + 9-s + 9·11-s − 3·12-s + 14·13-s + 3·15-s + 4·16-s + 12·17-s + 9·19-s − 9·20-s + 12·23-s − 3·25-s + 4·27-s + 9·29-s − 30·31-s − 9·33-s + 3·36-s − 14·39-s + 18·41-s − 5·43-s + 27·44-s − 3·45-s − 24·47-s − 4·48-s − 12·51-s + ⋯
 L(s)  = 1 − 0.577·3-s + 3/2·4-s − 1.34·5-s + 1/3·9-s + 2.71·11-s − 0.866·12-s + 3.88·13-s + 0.774·15-s + 16-s + 2.91·17-s + 2.06·19-s − 2.01·20-s + 2.50·23-s − 3/5·25-s + 0.769·27-s + 1.67·29-s − 5.38·31-s − 1.56·33-s + 1/2·36-s − 2.24·39-s + 2.81·41-s − 0.762·43-s + 4.07·44-s − 0.447·45-s − 3.50·47-s − 0.577·48-s − 1.68·51-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

 $$L(1)$$ $$\approx$$ $$6.323170502$$ $$L(\frac12)$$ $$\approx$$ $$6.323170502$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - 7 T + p T^{2} )^{2}$$
good2$C_2^2$$\times$$C_2^2$ $$( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} )$$
3$D_4\times C_2$ $$1 + T - 5 T^{3} - 11 T^{4} - 5 p T^{5} + p^{3} T^{7} + p^{4} T^{8}$$
5$D_4\times C_2$ $$1 + 3 T + 12 T^{2} + 27 T^{3} + 71 T^{4} + 27 p T^{5} + 12 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 9 T + 54 T^{2} - 243 T^{3} + 905 T^{4} - 243 p T^{5} + 54 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2$ $$( 1 - 3 T + p T^{2} )^{4}$$
19$D_4\times C_2$ $$1 - 9 T + 56 T^{2} - 261 T^{3} + 993 T^{4} - 261 p T^{5} + 56 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
23$D_{4}$ $$( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2$ $$( 1 + 4 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2}$$
41$D_4\times C_2$ $$1 - 18 T + 210 T^{2} - 1836 T^{3} + 13151 T^{4} - 1836 p T^{5} + 210 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 + 24 T + 327 T^{2} + 3240 T^{3} + 25040 T^{4} + 3240 p T^{5} + 327 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 6 T + 5 T^{2} + 450 T^{3} - 3756 T^{4} + 450 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 6 T^{2} + 5627 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 2 T + 70 T^{2} + 376 T^{3} + 391 T^{4} + 376 p T^{5} + 70 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2$$\times$$C_2^2$ $$( 1 + 4 T + p T^{2} )^{2}( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )$$
71$D_4\times C_2$ $$1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 87 T^{2} + 1853 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 + 39 T + 812 T^{2} + 11895 T^{3} + 132795 T^{4} + 11895 p T^{5} + 812 p^{2} T^{6} + 39 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$