# Properties

 Degree $8$ Conductor $6.045\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 2·5-s + 2·9-s + 4·11-s − 12·13-s − 4·15-s + 2·19-s − 8·23-s + 6·25-s + 8·27-s + 4·31-s − 8·33-s + 24·39-s + 16·41-s + 8·43-s + 4·45-s + 12·47-s + 20·53-s + 8·55-s − 4·57-s − 14·59-s + 18·61-s − 24·65-s − 8·67-s + 16·69-s + 16·71-s + 12·73-s + ⋯
 L(s)  = 1 − 1.15·3-s + 0.894·5-s + 2/3·9-s + 1.20·11-s − 3.32·13-s − 1.03·15-s + 0.458·19-s − 1.66·23-s + 6/5·25-s + 1.53·27-s + 0.718·31-s − 1.39·33-s + 3.84·39-s + 2.49·41-s + 1.21·43-s + 0.596·45-s + 1.75·47-s + 2.74·53-s + 1.07·55-s − 0.529·57-s − 1.82·59-s + 2.30·61-s − 2.97·65-s − 0.977·67-s + 1.92·69-s + 1.89·71-s + 1.40·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{20} \cdot 7^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1568} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{20} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.752374701$$ $$L(\frac12)$$ $$\approx$$ $$1.752374701$$ $$L(\frac{3}{2})$$ 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 $$1$$
7 $$1$$
good3$C_2^3$ $$1 + 2 T + 2 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
5$D_4\times C_2$ $$1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 8 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
11$C_4\times C_2$ $$1 - 4 T + 10 T^{2} + 64 T^{3} - 261 T^{4} + 64 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
13$D_{4}$ $$( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
17$C_2^2$$\times$$C_2^2$ $$( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )$$
19$D_4\times C_2$ $$1 - 2 T - 30 T^{2} + 8 T^{3} + 719 T^{4} + 8 p T^{5} - 30 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 38 T^{2} + p^{2} T^{4} )^{2}$$
31$C_4\times C_2$ $$1 - 4 T - 30 T^{2} + 64 T^{3} + 659 T^{4} + 64 p T^{5} - 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2^3$ $$1 - 54 T^{2} + 1547 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 12 T + 34 T^{2} - 192 T^{3} + 3123 T^{4} - 192 p T^{5} + 34 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 + 14 T + 34 T^{2} + 616 T^{3} + 11199 T^{4} + 616 p T^{5} + 34 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 18 T + 126 T^{2} - 1368 T^{3} + 15719 T^{4} - 1368 p T^{5} + 126 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^2$ $$( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_{4}$ $$( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 12 T + 42 T^{2} + 528 T^{3} - 5437 T^{4} + 528 p T^{5} + 42 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 + 8 T - 30 T^{2} - 512 T^{3} - 2461 T^{4} - 512 p T^{5} - 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_{4}$ $$( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$