# Properties

 Label 8-1400e4-1.1-c0e4-0-2 Degree $8$ Conductor $3.842\times 10^{12}$ Sign $1$ Analytic cond. $0.238309$ Root an. cond. $0.835877$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 3·3-s + 5-s + 3·6-s + 4·7-s + 6·9-s − 10-s + 2·13-s − 4·14-s − 3·15-s − 6·18-s + 2·19-s − 12·21-s − 2·23-s − 2·26-s − 10·27-s + 3·30-s + 32-s + 4·35-s − 2·38-s − 6·39-s + 12·42-s + 6·45-s + 2·46-s + 10·49-s + 10·54-s − 6·57-s + ⋯
 L(s)  = 1 − 2-s − 3·3-s + 5-s + 3·6-s + 4·7-s + 6·9-s − 10-s + 2·13-s − 4·14-s − 3·15-s − 6·18-s + 2·19-s − 12·21-s − 2·23-s − 2·26-s − 10·27-s + 3·30-s + 32-s + 4·35-s − 2·38-s − 6·39-s + 12·42-s + 6·45-s + 2·46-s + 10·49-s + 10·54-s − 6·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 5^{8} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$0.238309$$ Root analytic conductor: $$0.835877$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1400} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5360146764$$ $$L(\frac12)$$ $$\approx$$ $$0.5360146764$$ $$L(1)$$ 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$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
5$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
7$C_1$ $$( 1 - T )^{4}$$
good3$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
11$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
13$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
17$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
19$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
23$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
29$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
31$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
37$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
41$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
47$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
53$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
59$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
61$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
67$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
71$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
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$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
89$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
97$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$