# Properties

 Label 8-15e8-1.1-c1e4-0-3 Degree $8$ Conductor $2562890625$ Sign $1$ Analytic cond. $10.4192$ Root an. cond. $1.34038$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·4-s + 5·5-s − 2·7-s + 5·8-s + 10·10-s + 2·11-s + 9·13-s − 4·14-s + 5·16-s − 8·17-s − 5·19-s + 10·20-s + 4·22-s + 11·23-s + 10·25-s + 18·26-s − 4·28-s − 5·29-s + 3·31-s − 2·32-s − 16·34-s − 10·35-s − 7·37-s − 10·38-s + 25·40-s − 8·41-s + ⋯
 L(s)  = 1 + 1.41·2-s + 4-s + 2.23·5-s − 0.755·7-s + 1.76·8-s + 3.16·10-s + 0.603·11-s + 2.49·13-s − 1.06·14-s + 5/4·16-s − 1.94·17-s − 1.14·19-s + 2.23·20-s + 0.852·22-s + 2.29·23-s + 2·25-s + 3.53·26-s − 0.755·28-s − 0.928·29-s + 0.538·31-s − 0.353·32-s − 2.74·34-s − 1.69·35-s − 1.15·37-s − 1.62·38-s + 3.95·40-s − 1.24·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$10.4192$$ Root analytic conductor: $$1.34038$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$6.031943019$$ $$L(\frac12)$$ $$\approx$$ $$6.031943019$$ $$L(\frac{3}{2})$$ 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)$
bad3 $$1$$
5$C_4$ $$1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4}$$
good2$C_2^2:C_4$ $$1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8}$$
7$D_{4}$ $$( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
11$C_4\times C_2$ $$1 - 2 T + 13 T^{2} - 34 T^{3} + 225 T^{4} - 34 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2^2:C_4$ $$1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 15 p T^{5} + 23 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2^2:C_4$ $$1 + 8 T + 7 T^{2} - 110 T^{3} - 579 T^{4} - 110 p T^{5} + 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2^2:C_4$ $$1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2^2:C_4$ $$1 - 11 T + 28 T^{2} + 245 T^{3} - 2259 T^{4} + 245 p T^{5} + 28 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2^2:C_4$ $$1 + 5 T - 19 T^{2} - 5 p T^{3} - 4 T^{4} - 5 p^{2} T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
31$C_4\times C_2$ $$1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 159 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^2:C_4$ $$1 + 7 T - 18 T^{2} - 145 T^{3} + 371 T^{4} - 145 p T^{5} - 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2:C_4$ $$1 + 8 T - 17 T^{2} - 254 T^{3} - 435 T^{4} - 254 p T^{5} - 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2:C_4$ $$1 - 2 T - 43 T^{2} - 50 T^{3} + 2351 T^{4} - 50 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2:C_4$ $$1 + 9 T + 8 T^{2} + 315 T^{3} + 5131 T^{4} + 315 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2^2:C_4$ $$1 + 31 T^{2} - 210 T^{3} + 2851 T^{4} - 210 p T^{5} + 31 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2:C_4$ $$1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 941 p T^{5} + 78 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^2:C_4$ $$1 + 2 T - 3 T^{2} - 410 T^{3} + 1601 T^{4} - 410 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2^2:C_4$ $$1 + 8 T - 37 T^{2} - 694 T^{3} - 2425 T^{4} - 694 p T^{5} - 37 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
73$C_4\times C_2$ $$1 - 9 T + 8 T^{2} + 585 T^{3} - 5849 T^{4} + 585 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2^2:C_4$ $$1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2^2:C_4$ $$1 + 9 T - 52 T^{2} - 675 T^{3} + 121 T^{4} - 675 p T^{5} - 52 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}$$
89$C_4\times C_2$ $$1 - 20 T + 151 T^{2} - 1600 T^{3} + 21441 T^{4} - 1600 p T^{5} + 151 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2^2:C_4$ $$1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 20 p T^{5} - 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$