# Properties

 Degree 8 Conductor $3^{8} \cdot 7^{4} \cdot 11^{8}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 4

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 4·7-s − 8·13-s + 4·16-s + 8·19-s − 6·25-s − 16·28-s − 8·31-s − 16·37-s − 4·43-s + 10·49-s + 32·52-s − 8·61-s + 16·64-s + 4·67-s − 48·73-s − 32·76-s − 24·79-s − 32·91-s + 16·97-s + 24·100-s − 8·103-s + 36·109-s + 16·112-s + 32·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 − 2·4-s + 1.51·7-s − 2.21·13-s + 16-s + 1.83·19-s − 6/5·25-s − 3.02·28-s − 1.43·31-s − 2.63·37-s − 0.609·43-s + 10/7·49-s + 4.43·52-s − 1.02·61-s + 2·64-s + 0.488·67-s − 5.61·73-s − 3.67·76-s − 2.70·79-s − 3.35·91-s + 1.62·97-s + 12/5·100-s − 0.788·103-s + 3.44·109-s + 1.51·112-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$3^{8} \cdot 7^{4} \cdot 11^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{7623} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 4 Selberg data = $(8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;7,\;11\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
7$C_1$ $$( 1 - T )^{4}$$
11 $$1$$
good2$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
5$C_2^2$ $$( 1 + 3 T^{2} + p^{2} T^{4} )^{2}$$
13$D_{4}$ $$( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 + 38 T^{2} + 715 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8}$$
19$D_{4}$ $$( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 + 32 T^{2} + 1090 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 + 56 T^{2} + 2242 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8}$$
31$D_{4}$ $$( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_{4}$ $$( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + T + p T^{2} )^{4}$$
47$D_4\times C_2$ $$1 + 158 T^{2} + 10435 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 28 T^{2} + 2230 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 + 158 T^{2} + 12307 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8}$$
61$D_{4}$ $$( 1 + 4 T + 112 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_{4}$ $$( 1 - 2 T + 79 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 140 T^{2} + p^{2} T^{4} )^{2}$$
73$D_{4}$ $$( 1 + 24 T + 276 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_{4}$ $$( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 + 174 T^{2} + 19331 T^{4} + 174 p^{2} T^{6} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 + 198 T^{2} + 23627 T^{4} + 198 p^{2} T^{6} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}