# Properties

 Label 8-725e4-1.1-c1e4-0-5 Degree $8$ Conductor $276281640625$ Sign $1$ Analytic cond. $1123.20$ Root an. cond. $2.40606$ 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 + 2·4-s − 5·5-s − 12·7-s + 13·9-s + 4·11-s − 8·12-s + 13-s + 20·15-s + 4·17-s + 17·19-s − 10·20-s + 48·21-s + 18·23-s + 10·25-s − 30·27-s − 24·28-s − 29-s + 9·31-s − 16·33-s + 60·35-s + 26·36-s + 22·37-s − 4·39-s − 2·41-s + 8·44-s − 65·45-s + ⋯
 L(s)  = 1 − 2.30·3-s + 4-s − 2.23·5-s − 4.53·7-s + 13/3·9-s + 1.20·11-s − 2.30·12-s + 0.277·13-s + 5.16·15-s + 0.970·17-s + 3.90·19-s − 2.23·20-s + 10.4·21-s + 3.75·23-s + 2·25-s − 5.77·27-s − 4.53·28-s − 0.185·29-s + 1.61·31-s − 2.78·33-s + 10.1·35-s + 13/3·36-s + 3.61·37-s − 0.640·39-s − 0.312·41-s + 1.20·44-s − 9.68·45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5^{8} \cdot 29^{4}$$ Sign: $1$ Analytic conductor: $$1123.20$$ Root analytic conductor: $$2.40606$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5^{8} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9855310958$$ $$L(\frac12)$$ $$\approx$$ $$0.9855310958$$ $$L(\frac{3}{2})$$ 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)$
bad5$C_4$ $$1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
29$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
good2$C_4\times C_2$ $$1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8}$$
3$C_2^2:C_4$ $$1 + 4 T + p T^{2} - 10 T^{3} - 29 T^{4} - 10 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
7$C_2$ $$( 1 + 3 T + p T^{2} )^{4}$$
11$C_2^2:C_4$ $$1 - 4 T - 5 T^{2} + 34 T^{3} - 21 T^{4} + 34 p T^{5} - 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
13$C_4\times C_2$ $$1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 25 p T^{5} - 12 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
17$C_2^2:C_4$ $$1 - 4 T - T^{2} - 58 T^{3} + 509 T^{4} - 58 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2^2:C_4$ $$1 - 17 T + 165 T^{2} - 1117 T^{3} + 5564 T^{4} - 1117 p T^{5} + 165 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2^2:C_4$ $$1 - 18 T + 7 p T^{2} - 1014 T^{3} + 5239 T^{4} - 1014 p T^{5} + 7 p^{3} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2^2:C_4$ $$1 - 9 T + 75 T^{2} - 521 T^{3} + 3864 T^{4} - 521 p T^{5} + 75 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^2:C_4$ $$1 - 22 T + 207 T^{2} - 1220 T^{3} + 6701 T^{4} - 1220 p T^{5} + 207 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2:C_4$ $$1 + 2 T - 37 T^{2} + 44 T^{3} + 1805 T^{4} + 44 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2^2$ $$( 1 + 41 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^2:C_4$ $$1 - 6 T - 11 T^{2} - 222 T^{3} + 3559 T^{4} - 222 p T^{5} - 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2:C_4$ $$1 + 12 T + 91 T^{2} + 966 T^{3} + 9829 T^{4} + 966 p T^{5} + 91 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2^2:C_4$ $$1 - 5 T + 56 T^{2} - 245 T^{3} + 1001 T^{4} - 245 p T^{5} + 56 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2^2:C_4$ $$1 - 25 T + 399 T^{2} - 4655 T^{3} + 41216 T^{4} - 4655 p T^{5} + 399 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^2:C_4$ $$1 - 5 T + 48 T^{2} - 85 T^{3} + 449 T^{4} - 85 p T^{5} + 48 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2^2:C_4$ $$1 - 19 T + 115 T^{2} - 731 T^{3} + 8664 T^{4} - 731 p T^{5} + 115 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2:C_4$ $$1 + 12 T - 19 T^{2} - 684 T^{3} - 3131 T^{4} - 684 p T^{5} - 19 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2^2:C_4$ $$1 + 15 T + 186 T^{2} + 2165 T^{3} + 25461 T^{4} + 2165 p T^{5} + 186 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
83$C_4\times C_2$ $$1 - 9 T - 2 T^{2} + 765 T^{3} - 6719 T^{4} + 765 p T^{5} - 2 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
89$C_4\times C_2$ $$1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 200 p T^{5} - 29 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
97$C_4\times C_2$ $$1 + 3 T - 88 T^{2} - 555 T^{3} + 6871 T^{4} - 555 p T^{5} - 88 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$