# Properties

 Degree 8 Conductor $3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s − 4·4-s + 6·9-s − 12·11-s − 16·12-s + 4·16-s − 12·17-s − 2·25-s − 4·27-s + 12·29-s + 16·31-s − 48·33-s − 24·36-s + 16·37-s − 24·41-s + 48·44-s + 16·48-s − 2·49-s − 48·51-s + 16·64-s − 16·67-s + 48·68-s − 8·75-s − 37·81-s + 12·83-s + 48·87-s + 64·93-s + ⋯
 L(s)  = 1 + 2.30·3-s − 2·4-s + 2·9-s − 3.61·11-s − 4.61·12-s + 16-s − 2.91·17-s − 2/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s − 8.35·33-s − 4·36-s + 2.63·37-s − 3.74·41-s + 7.23·44-s + 2.30·48-s − 2/7·49-s − 6.72·51-s + 2·64-s − 1.95·67-s + 5.82·68-s − 0.923·75-s − 4.11·81-s + 1.31·83-s + 5.14·87-s + 6.63·93-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{1155} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$0.7340658765$$ $$L(\frac12)$$ $$\approx$$ $$0.7340658765$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7,\;11\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
5$C_2$ $$( 1 + T^{2} )^{2}$$
7$C_2$ $$( 1 + T^{2} )^{2}$$
11$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
good2$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
17$D_{4}$ $$( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 2 p T^{2} + 1011 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 38 T^{2} + 771 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_{4}$ $$( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_{4}$ $$( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
41$D_{4}$ $$( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 134 T^{2} + 8115 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 16 T^{2} - 2718 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 190 T^{2} + 14571 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 22 T^{2} + 3555 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 158 T^{2} + 11883 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 272 T^{2} + 30690 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8}$$
83$D_{4}$ $$( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 98 T^{2} + 16443 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2$ $$( 1 + 7 T + p T^{2} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}