# Properties

 Degree 8 Conductor $2^{20} \cdot 3^{8} \cdot 7^{4}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 16·11-s − 72·17-s − 8·19-s + 16·25-s + 40·41-s + 80·43-s − 14·49-s + 184·59-s + 224·67-s + 232·73-s + 88·83-s − 312·89-s − 136·97-s − 96·107-s + 304·113-s − 180·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 592·169-s + 173-s + ⋯
 L(s)  = 1 + 1.45·11-s − 4.23·17-s − 0.421·19-s + 0.639·25-s + 0.975·41-s + 1.86·43-s − 2/7·49-s + 3.11·59-s + 3.34·67-s + 3.17·73-s + 1.06·83-s − 3.50·89-s − 1.40·97-s − 0.897·107-s + 2.69·113-s − 1.48·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.50·169-s + 0.00578·173-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{20} \cdot 3^{8} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{2016} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$4.100479064$$ $$L(\frac12)$$ $$\approx$$ $$4.100479064$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
good5$D_4\times C_2$ $$1 - 16 T^{2} - 254 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8}$$
11$D_{4}$ $$( 1 - 8 T + 186 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 592 T^{2} + 143170 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8}$$
17$D_{4}$ $$( 1 + 36 T + 870 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
19$D_{4}$ $$( 1 + 4 T + 4 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 268 T^{2} + 477286 T^{4} - 268 p^{4} T^{6} + p^{8} T^{8}$$
29$C_2^2$ $$( 1 - 1178 T^{2} + p^{4} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 1380 T^{2} + 1419974 T^{4} - 1380 p^{4} T^{6} + p^{8} T^{8}$$
37$D_4\times C_2$ $$1 - 1780 T^{2} + 2031622 T^{4} - 1780 p^{4} T^{6} + p^{8} T^{8}$$
41$D_{4}$ $$( 1 - 20 T + 3174 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 40 T + 4090 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 7492 T^{2} + 23390470 T^{4} - 7492 p^{4} T^{6} + p^{8} T^{8}$$
53$D_4\times C_2$ $$1 - 1604 T^{2} - 6155034 T^{4} - 1604 p^{4} T^{6} + p^{8} T^{8}$$
59$D_{4}$ $$( 1 - 92 T + 8836 T^{2} - 92 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 - 13232 T^{2} + 71110338 T^{4} - 13232 p^{4} T^{6} + p^{8} T^{8}$$
67$D_{4}$ $$( 1 - 112 T + 11602 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 9412 T^{2} + 47279686 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 - 116 T + 13894 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 - 16900 T^{2} + 142880134 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8}$$
83$D_{4}$ $$( 1 - 44 T + 13924 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
89$D_{4}$ $$( 1 + 156 T + 20774 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
97$D_{4}$ $$( 1 + 68 T + 3046 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}