# Properties

 Label 8-1323e4-1.1-c1e4-0-4 Degree $8$ Conductor $3.064\times 10^{12}$ Sign $1$ Analytic cond. $12455.1$ Root an. cond. $3.25026$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 8·5-s − 20·17-s + 16·20-s + 30·25-s + 8·37-s − 8·41-s − 12·47-s − 20·59-s − 36·67-s + 40·68-s − 28·79-s − 8·83-s + 160·85-s − 20·89-s − 60·100-s − 28·101-s + 48·109-s − 30·121-s − 80·125-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 4-s − 3.57·5-s − 4.85·17-s + 3.57·20-s + 6·25-s + 1.31·37-s − 1.24·41-s − 1.75·47-s − 2.60·59-s − 4.39·67-s + 4.85·68-s − 3.15·79-s − 0.878·83-s + 17.3·85-s − 2.11·89-s − 6·100-s − 2.78·101-s + 4.59·109-s − 2.72·121-s − 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \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(3^{12} \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: $$3^{12} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$12455.1$$ Root analytic conductor: $$3.25026$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1323} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 $$1$$
7 $$1$$
good2$C_4\times C_2$ $$1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8}$$
5$D_{4}$ $$( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 + 30 T^{2} + 422 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2$ $$( 1 + 5 T + p T^{2} )^{4}$$
19$D_4\times C_2$ $$1 + 30 T^{2} + 902 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 56 T^{2} + 1522 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 - 10 T^{2} + 1582 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 + 70 T^{2} + 2902 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8}$$
37$D_{4}$ $$( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$D_{4}$ $$( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 + 81 T^{2} + p^{2} T^{4} )^{2}$$
47$D_{4}$ $$( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 96 T^{2} + p^{2} T^{4} )^{2}$$
59$D_{4}$ $$( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 + 230 T^{2} + 20622 T^{4} + 230 p^{2} T^{6} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 + 68 T^{2} + 7318 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 + 112 T^{2} + 5794 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 + 14 T + 187 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_{4}$ $$( 1 + 4 T + 165 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_{4}$ $$( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 + 358 T^{2} + 50734 T^{4} + 358 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$