# Properties

 Degree $8$ Conductor $1.642\times 10^{13}$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·2-s − 3-s + 6·4-s + 3·6-s − 10·8-s + 11-s − 6·12-s + 3·13-s + 15·16-s + 2·17-s + 3·19-s − 3·22-s + 2·23-s + 10·24-s − 25-s − 9·26-s + 2·29-s − 22·32-s − 33-s − 6·34-s − 9·38-s − 3·39-s + 6·44-s − 6·46-s − 15·48-s − 49-s + 3·50-s + ⋯
 L(s)  = 1 − 3·2-s − 3-s + 6·4-s + 3·6-s − 10·8-s + 11-s − 6·12-s + 3·13-s + 15·16-s + 2·17-s + 3·19-s − 3·22-s + 2·23-s + 10·24-s − 25-s − 9·26-s + 2·29-s − 22·32-s − 33-s − 6·34-s − 9·38-s − 3·39-s + 6·44-s − 6·46-s − 15·48-s − 49-s + 3·50-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 11^{4} \cdot 61^{4}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{2013} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 11^{4} \cdot 61^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4124570705$$ $$L(\frac12)$$ $$\approx$$ $$0.4124570705$$ $$L(1)$$ 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$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
11$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
61$C_4$ $$1 + T + T^{2} + T^{3} + T^{4}$$
good2$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
5$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
7$C_4$$\times$$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
13$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
17$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
19$C_1$$\times$$C_4$ $$( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
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_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_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
59$C_1$$\times$$C_4$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
67$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
71$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
73$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
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$C_4$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
97$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$