# Properties

 Label 8-2e40-1.1-c1e4-0-10 Degree $8$ Conductor $1.100\times 10^{12}$ Sign $1$ Analytic cond. $4470.00$ Root an. cond. $2.85948$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·5-s − 4·9-s + 8·13-s + 24·25-s + 24·29-s + 8·37-s − 16·41-s − 32·45-s − 12·49-s + 24·53-s + 24·61-s + 64·65-s + 8·73-s + 2·81-s + 8·89-s + 32·97-s + 40·101-s − 24·109-s − 8·113-s − 32·117-s − 4·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 192·145-s + ⋯
 L(s)  = 1 + 3.57·5-s − 4/3·9-s + 2.21·13-s + 24/5·25-s + 4.45·29-s + 1.31·37-s − 2.49·41-s − 4.77·45-s − 1.71·49-s + 3.29·53-s + 3.07·61-s + 7.93·65-s + 0.936·73-s + 2/9·81-s + 0.847·89-s + 3.24·97-s + 3.98·101-s − 2.29·109-s − 0.752·113-s − 2.95·117-s − 0.363·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.9·145-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{40}$$ Sign: $1$ Analytic conductor: $$4470.00$$ Root analytic conductor: $$2.85948$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1024} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{40} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$10.22171440$$ $$L(\frac12)$$ $$\approx$$ $$10.22171440$$ $$L(\frac{3}{2})$$ 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 $$1$$
good3$C_2^2:C_4$ $$1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}$$
5$D_{4}$ $$( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
7$C_2^2:C_4$ $$1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2:C_4$ $$1 + 4 T^{2} + 238 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}$$
13$D_{4}$ $$( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + 26 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2:C_4$ $$1 + 36 T^{2} + 654 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2^2:C_4$ $$1 + 12 T^{2} + 1062 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2:C_4$ $$1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8}$$
37$D_{4}$ $$( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
43$C_2^2:C_4$ $$1 + 164 T^{2} + 10414 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8}$$
47$C_2^2:C_4$ $$1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8}$$
53$D_{4}$ $$( 1 - 12 T + 140 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2:C_4$ $$1 + 228 T^{2} + 19950 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8}$$
61$D_{4}$ $$( 1 - 12 T + 108 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2:C_4$ $$1 + 164 T^{2} + 13390 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2^2:C_4$ $$1 + 76 T^{2} + 9958 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8}$$
73$D_{4}$ $$( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^2:C_4$ $$1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8}$$
83$C_2^2:C_4$ $$1 + 132 T^{2} + 10446 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_{4}$ $$( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$