# Properties

 Label 8-1792e4-1.1-c1e4-0-19 Degree $8$ Conductor $1.031\times 10^{13}$ Sign $1$ Analytic cond. $41923.7$ Root an. cond. $3.78274$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·7-s − 4·9-s − 8·17-s − 16·23-s − 12·25-s − 8·41-s − 32·47-s + 10·49-s + 16·63-s − 32·71-s − 24·73-s − 32·79-s + 2·81-s + 8·89-s − 40·97-s − 32·103-s + 24·113-s + 32·119-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s + ⋯
 L(s)  = 1 − 1.51·7-s − 4/3·9-s − 1.94·17-s − 3.33·23-s − 2.39·25-s − 1.24·41-s − 4.66·47-s + 10/7·49-s + 2.01·63-s − 3.79·71-s − 2.80·73-s − 3.60·79-s + 2/9·81-s + 0.847·89-s − 4.06·97-s − 3.15·103-s + 2.25·113-s + 2.93·119-s − 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.58·153-s + 0.0798·157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{32} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$41923.7$$ Root analytic conductor: $$3.78274$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1792} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## 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)$
bad2 $$1$$
7$C_1$ $$( 1 + T )^{4}$$
good3$C_2^2:C_4$ $$1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8}$$
5$C_2^2:C_4$ $$1 + 12 T^{2} + 78 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2:C_4$ $$1 + 12 T^{2} + 150 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2^2:C_4$ $$1 + 12 T^{2} - 18 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8}$$
17$C_4$ $$( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
19$C_2^2:C_4$ $$1 + 36 T^{2} + 1038 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8}$$
23$D_{4}$ $$( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2^2:C_4$ $$1 - 44 T^{2} + 2038 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}$$
31$C_2^2$ $$( 1 + 30 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2:C_4$ $$1 + 116 T^{2} + 5974 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2^2:C_4$ $$1 + 140 T^{2} + 8470 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8}$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
53$C_2^2:C_4$ $$1 + 84 T^{2} + 5334 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2:C_4$ $$1 + 196 T^{2} + 16174 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2:C_4$ $$1 + 44 T^{2} + 5614 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8}$$
67$C_2^2:C_4$ $$1 + 204 T^{2} + 18870 T^{4} + 204 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_{4}$ $$( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_{4}$ $$( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2:C_4$ $$1 + 132 T^{2} + 13134 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
97$C_2$ $$( 1 + 10 T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$