# Properties

 Label 8-171e4-1.1-c0e4-0-0 Degree $8$ Conductor $855036081$ Sign $1$ Analytic cond. $5.30411\times 10^{-5}$ Root an. cond. $0.292130$ 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 − 2·5-s − 2·7-s − 2·9-s + 2·11-s + 16-s + 2·20-s + 2·23-s + 3·25-s + 2·28-s + 4·35-s + 2·36-s − 2·43-s − 2·44-s + 4·45-s − 2·47-s + 3·49-s − 4·55-s + 2·61-s + 4·63-s − 2·64-s − 4·77-s − 2·80-s + 3·81-s + 2·83-s − 2·92-s − 4·99-s + ⋯
 L(s)  = 1 − 4-s − 2·5-s − 2·7-s − 2·9-s + 2·11-s + 16-s + 2·20-s + 2·23-s + 3·25-s + 2·28-s + 4·35-s + 2·36-s − 2·43-s − 2·44-s + 4·45-s − 2·47-s + 3·49-s − 4·55-s + 2·61-s + 4·63-s − 2·64-s − 4·77-s − 2·80-s + 3·81-s + 2·83-s − 2·92-s − 4·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \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(3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$5.30411\times 10^{-5}$$ Root analytic conductor: $$0.292130$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1025405157$$ $$L(\frac12)$$ $$\approx$$ $$0.1025405157$$ $$L(1)$$ 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)$
bad3$C_2$ $$( 1 + T^{2} )^{2}$$
19$C_2$ $$( 1 + T^{2} )^{2}$$
good2$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
5$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
7$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
11$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
13$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
17$C_2$ $$( 1 + T^{2} )^{4}$$
23$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
29$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
31$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
37$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
41$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
43$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
47$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
59$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
61$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
67$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
73$C_2$ $$( 1 + T^{2} )^{4}$$
79$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
83$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
97$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$