# Properties

 Label 8-2e36-1.1-c1e4-0-6 Degree $8$ Conductor $68719476736$ Sign $1$ Analytic cond. $279.375$ Root an. cond. $2.02196$ 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 + 8·5-s − 4·7-s + 10·9-s + 4·11-s + 32·15-s + 12·19-s − 16·21-s − 12·23-s + 30·25-s + 20·27-s + 8·29-s − 16·31-s + 16·33-s − 32·35-s + 12·41-s + 12·43-s + 80·45-s + 8·49-s + 16·53-s + 32·55-s + 48·57-s − 4·59-s − 40·63-s + 12·67-s − 48·69-s + 12·71-s + ⋯
 L(s)  = 1 + 2.30·3-s + 3.57·5-s − 1.51·7-s + 10/3·9-s + 1.20·11-s + 8.26·15-s + 2.75·19-s − 3.49·21-s − 2.50·23-s + 6·25-s + 3.84·27-s + 1.48·29-s − 2.87·31-s + 2.78·33-s − 5.40·35-s + 1.87·41-s + 1.82·43-s + 11.9·45-s + 8/7·49-s + 2.19·53-s + 4.31·55-s + 6.35·57-s − 0.520·59-s − 5.03·63-s + 1.46·67-s − 5.77·69-s + 1.42·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{36}$$ Sign: $1$ Analytic conductor: $$279.375$$ Root analytic conductor: $$2.02196$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{512} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{36} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$14.24038468$$ $$L(\frac12)$$ $$\approx$$ $$14.24038468$$ $$L(\frac{3}{2})$$ 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$$\times$$C_2^2$ $$( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 - 2 T^{2} + p^{2} T^{4} )$$
5$C_2$$\times$$C_2^2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 + 8 T^{2} + p^{2} T^{4} )$$
7$C_2^2$ $$( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2^2$$\times$$C_2^2$ $$( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T^{2} + p^{2} T^{4} )$$
13$D_4\times C_2$ $$1 + 2 T^{2} + 48 T^{3} + 2 T^{4} + 48 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2^2$ $$( 1 - 26 T^{2} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 12 T + 86 T^{2} - 492 T^{3} + 2402 T^{4} - 492 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 - 8 T + 34 T^{2} - 200 T^{3} + 1026 T^{4} - 200 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
37$D_4\times C_2$ $$1 + 2 T^{2} + 144 T^{3} + 2 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 - 12 T + 72 T^{2} - 516 T^{3} + 3694 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 12 T + 38 T^{2} + 564 T^{3} - 6334 T^{4} + 564 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 16 T + 82 T^{2} + 128 T^{3} - 3294 T^{4} + 128 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + 4 T + 22 T^{2} - 668 T^{3} - 2718 T^{4} - 668 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 2 T^{2} + 240 T^{3} + 2 T^{4} + 240 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 12 T + 86 T^{2} - 876 T^{3} + 7010 T^{4} - 876 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 876 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 122 T^{2} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 4 T + 22 T^{2} + 1052 T^{3} - 4254 T^{4} + 1052 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$