# Properties

 Degree 8 Conductor $3^{4} \cdot 11^{4} \cdot 61^{4}$ 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

 $$d$$ = $$8$$ $$N$$ = $$3^{4} \cdot 11^{4} \cdot 61^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{2013} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 3^{4} \cdot 11^{4} \cdot 61^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$ $$L(\frac{1}{2})$$ $$\approx$$ $$8.691883791$$ $$L(\frac12)$$ $$\approx$$ $$8.691883791$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;11,\;61\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{3,\;11,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$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$C_4$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}