# Properties

 Degree 8 Conductor $2^{16} \cdot 3^{8}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 4

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 6·3-s + 2·4-s − 8·5-s + 12·6-s − 6·7-s − 4·8-s + 21·9-s + 16·10-s − 10·11-s − 12·12-s − 10·13-s + 12·14-s + 48·15-s + 8·16-s − 16·17-s − 42·18-s − 6·19-s − 16·20-s + 36·21-s + 20·22-s + 6·23-s + 24·24-s + 44·25-s + 20·26-s − 54·27-s − 12·28-s + ⋯
 L(s)  = 1 − 1.41·2-s − 3.46·3-s + 4-s − 3.57·5-s + 4.89·6-s − 2.26·7-s − 1.41·8-s + 7·9-s + 5.05·10-s − 3.01·11-s − 3.46·12-s − 2.77·13-s + 3.20·14-s + 12.3·15-s + 2·16-s − 3.88·17-s − 9.89·18-s − 1.37·19-s − 3.57·20-s + 7.85·21-s + 4.26·22-s + 1.25·23-s + 4.89·24-s + 44/5·25-s + 3.92·26-s − 10.3·27-s − 2.26·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{16} \cdot 3^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{144} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$4$$ Selberg data = $$(8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$C_2$ $$( 1 + p T + p T^{2} )^{2}$$
good5$D_4\times C_2$ $$1 + 8 T + 4 p T^{2} - 4 T^{3} - 89 T^{4} - 4 p T^{5} + 4 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
7$D_4\times C_2$ $$1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 82 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2$$\times$$C_2^2$ $$( 1 + 5 T + p T^{2} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} )$$
17$C_2^2$ $$( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 132 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2^3$ $$1 + 6 T + 18 T^{2} + 36 T^{3} - 457 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 32 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^3$ $$1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 + 16 T + 65 T^{2} - 624 T^{3} - 8092 T^{4} - 624 p T^{5} + 65 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 148 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 976 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 6 T + 45 T^{2} + 594 T^{3} - 3376 T^{4} + 594 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 1596 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - 16 T + 113 T^{2} - 384 T^{3} - 172 T^{4} - 384 p T^{5} + 113 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^2$ $$( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^3$ $$1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 328 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 - 174 T^{2} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}