# Properties

 Label 8-490e4-1.1-c1e4-0-1 Degree $8$ Conductor $57648010000$ Sign $1$ Analytic cond. $234.364$ Root an. cond. $1.97804$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 4·5-s + 3·16-s + 16·19-s + 8·20-s + 8·25-s − 8·29-s − 16·31-s + 24·41-s − 16·59-s − 24·61-s − 4·64-s − 24·71-s − 32·76-s − 8·79-s − 12·80-s − 18·81-s − 40·89-s − 64·95-s − 16·100-s − 24·101-s − 8·109-s + 16·116-s + 4·121-s + 32·124-s − 20·125-s + 127-s + ⋯
 L(s)  = 1 − 4-s − 1.78·5-s + 3/4·16-s + 3.67·19-s + 1.78·20-s + 8/5·25-s − 1.48·29-s − 2.87·31-s + 3.74·41-s − 2.08·59-s − 3.07·61-s − 1/2·64-s − 2.84·71-s − 3.67·76-s − 0.900·79-s − 1.34·80-s − 2·81-s − 4.23·89-s − 6.56·95-s − 8/5·100-s − 2.38·101-s − 0.766·109-s + 1.48·116-s + 4/11·121-s + 2.87·124-s − 1.78·125-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\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 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 5^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$234.364$$ Root analytic conductor: $$1.97804$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{490} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.1397390913$$ $$L(\frac12)$$ $$\approx$$ $$0.1397390913$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
5$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
7 $$1$$
good3$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}$$
19$D_{4}$ $$( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}$$
41$D_{4}$ $$( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_{4}$ $$( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
71$D_{4}$ $$( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 236 T^{2} + 24198 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 160 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 + 10 T + p T^{2} )^{4}$$
97$D_4\times C_2$ $$1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$