# Properties

 Label 8-15e8-1.1-c1e4-0-0 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-s + 2·4-s − 5·5-s − 5·10-s + 6·11-s − 2·13-s + 8·17-s − 4·19-s − 10·20-s + 6·22-s − 10·23-s + 10·25-s − 2·26-s + 8·29-s − 11·32-s + 8·34-s − 15·37-s − 4·38-s + 4·43-s + 12·44-s − 10·46-s + 2·47-s + 12·49-s + 10·50-s − 4·52-s + 5·53-s − 30·55-s + ⋯
 L(s)  = 1 + 0.707·2-s + 4-s − 2.23·5-s − 1.58·10-s + 1.80·11-s − 0.554·13-s + 1.94·17-s − 0.917·19-s − 2.23·20-s + 1.27·22-s − 2.08·23-s + 2·25-s − 0.392·26-s + 1.48·29-s − 1.94·32-s + 1.37·34-s − 2.46·37-s − 0.648·38-s + 0.609·43-s + 1.80·44-s − 1.47·46-s + 0.291·47-s + 12/7·49-s + 1.41·50-s − 0.554·52-s + 0.686·53-s − 4.04·55-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

 $$L(1)$$ $$\approx$$ $$1.529367283$$ $$L(\frac12)$$ $$\approx$$ $$1.529367283$$ $$L(\frac{3}{2})$$ 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$C_4\times C_2$ $$1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
7$C_2^2$ $$( 1 - 6 T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^2:C_4$ $$1 - 6 T + 5 T^{2} + 6 T^{3} + 49 T^{4} + 6 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2^2:C_4$ $$1 + 2 T + 11 T^{2} + 16 T^{3} + 49 T^{4} + 16 p T^{5} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2^2:C_4$ $$1 - 8 T + p T^{2} - 60 T^{3} + 461 T^{4} - 60 p T^{5} + p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2^2:C_4$ $$1 + 4 T - 3 T^{2} + 62 T^{3} + 605 T^{4} + 62 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
23$C_4\times C_2$ $$1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 200 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2^2:C_4$ $$1 - 8 T + 5 T^{2} + 8 p T^{3} - 59 p T^{4} + 8 p^{2} T^{5} + 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2^2:C_4$ $$1 + 9 T^{2} - 110 T^{3} + 741 T^{4} - 110 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2^2:C_4$ $$1 + 15 T + 63 T^{2} + 65 T^{3} + 144 T^{4} + 65 p T^{5} + 63 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2:C_4$ $$1 - 31 T^{2} - 180 T^{3} + 1501 T^{4} - 180 p T^{5} - 31 p^{2} T^{6} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2:C_4$ $$1 - 2 T - 23 T^{2} - 250 T^{3} + 2601 T^{4} - 250 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2:C_4$ $$1 - 5 T - 43 T^{2} + 5 p T^{3} + 1244 T^{4} + 5 p^{2} T^{5} - 43 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
59$C_4\times C_2$ $$1 + 4 T - 43 T^{2} - 408 T^{3} + 905 T^{4} - 408 p T^{5} - 43 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2^2:C_4$ $$1 - 2 T - 57 T^{2} - 64 T^{3} + 3905 T^{4} - 64 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^2:C_4$ $$1 - 2 T - 43 T^{2} - 370 T^{3} + 5041 T^{4} - 370 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2^2:C_4$ $$1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2:C_4$ $$1 + 10 T + 27 T^{2} + 740 T^{3} + 11429 T^{4} + 740 p T^{5} + 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
79$C_4\times C_2$ $$1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8}$$
83$C_2^2:C_4$ $$1 - 18 T + 41 T^{2} + 1416 T^{3} - 18731 T^{4} + 1416 p T^{5} + 41 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2^2:C_4$ $$1 - 9 T - 43 T^{2} + 993 T^{3} - 4460 T^{4} + 993 p T^{5} - 43 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2^2:C_4$ $$1 - 2 T - 33 T^{2} - 820 T^{3} + 10061 T^{4} - 820 p T^{5} - 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$