# Properties

 Label 8-1232e4-1.1-c0e4-0-0 Degree $8$ Conductor $2.304\times 10^{12}$ Sign $1$ Analytic cond. $0.142912$ Root an. cond. $0.784122$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 7-s − 9-s + 11-s + 2·23-s − 25-s − 2·29-s + 3·37-s + 2·43-s + 3·53-s − 63-s + 2·67-s − 3·71-s + 77-s − 3·79-s − 99-s − 3·107-s − 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·161-s + 163-s + ⋯
 L(s)  = 1 + 7-s − 9-s + 11-s + 2·23-s − 25-s − 2·29-s + 3·37-s + 2·43-s + 3·53-s − 63-s + 2·67-s − 3·71-s + 77-s − 3·79-s − 99-s − 3·107-s − 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·161-s + 163-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 7^{4} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$0.142912$$ Root analytic conductor: $$0.784122$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1232} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.196055532$$ $$L(\frac12)$$ $$\approx$$ $$1.196055532$$ $$L(1)$$ 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$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
11$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
good3$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
5$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
13$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
17$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
19$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
23$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
29$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
31$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
37$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
41$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
43$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
47$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
53$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
59$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
61$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
67$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
71$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
73$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
79$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
83$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
97$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$