# Properties

 Label 8-1045e4-1.1-c0e4-0-3 Degree $8$ Conductor $1.193\times 10^{12}$ Sign $1$ Analytic cond. $0.0739764$ Root an. cond. $0.722165$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s − 5-s + 5·7-s + 9-s + 11-s − 5·17-s − 19-s − 20-s + 5·28-s − 5·35-s + 36-s + 44-s − 45-s + 14·49-s − 55-s − 3·61-s + 5·63-s − 5·68-s − 76-s + 5·77-s + 5·85-s + 95-s + 99-s + 2·101-s − 25·119-s + 127-s + 131-s + ⋯
 L(s)  = 1 + 4-s − 5-s + 5·7-s + 9-s + 11-s − 5·17-s − 19-s − 20-s + 5·28-s − 5·35-s + 36-s + 44-s − 45-s + 14·49-s − 55-s − 3·61-s + 5·63-s − 5·68-s − 76-s + 5·77-s + 5·85-s + 95-s + 99-s + 2·101-s − 25·119-s + 127-s + 131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5^{4} \cdot 11^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$0.0739764$$ Root analytic conductor: $$0.722165$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5^{4} \cdot 11^{4} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.804113662$$ $$L(\frac12)$$ $$\approx$$ $$1.804113662$$ $$L(1)$$ 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)$
bad5$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
11$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
19$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
good2$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
3$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
7$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
13$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
17$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
23$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
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$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
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$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
47$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
53$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
59$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
61$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
67$C_2$ $$( 1 + T^{2} )^{4}$$
71$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
73$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
79$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
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$C_4\times C_2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$