# Properties

 Label 8-1792e4-1.1-c1e4-0-18 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 − 4·3-s − 8·7-s + 4·9-s − 12·19-s + 32·21-s + 12·25-s + 4·27-s − 16·37-s + 34·49-s + 48·57-s − 36·59-s − 32·63-s − 48·75-s − 10·81-s − 36·83-s − 48·103-s + 64·111-s − 24·113-s + 12·121-s + 127-s + 131-s + 96·133-s + 137-s + 139-s − 136·147-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 2.30·3-s − 3.02·7-s + 4/3·9-s − 2.75·19-s + 6.98·21-s + 12/5·25-s + 0.769·27-s − 2.63·37-s + 34/7·49-s + 6.35·57-s − 4.68·59-s − 4.03·63-s − 5.54·75-s − 1.11·81-s − 3.95·83-s − 4.72·103-s + 6.07·111-s − 2.25·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s − 11.2·147-s + 0.0819·149-s + 0.0813·151-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$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
good3$D_{4}$ $$( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
5$D_4\times C_2$ $$1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 - 28 T^{2} + 426 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2^2$ $$( 1 - 18 T^{2} + p^{2} T^{4} )^{2}$$
19$D_{4}$ $$( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + 10 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 14 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
41$D_4\times C_2$ $$1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}$$
47$C_2^2$ $$( 1 + 46 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 58 T^{2} + p^{2} T^{4} )^{2}$$
59$D_{4}$ $$( 1 + 18 T + 172 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 - 76 T^{2} + 6186 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 172 T^{2} + 14646 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 - 98 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 146 T^{2} + p^{2} T^{4} )^{2}$$
83$D_{4}$ $$( 1 + 18 T + 244 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$