# Properties

 Degree 8 Conductor $43^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·2-s + 3·3-s + 17·4-s + 2·5-s − 18·6-s + 4·7-s − 30·8-s + 7·9-s − 12·10-s − 10·11-s + 51·12-s − 5·13-s − 24·14-s + 6·15-s + 40·16-s − 17-s − 42·18-s − 2·19-s + 34·20-s + 12·21-s + 60·22-s − 11·23-s − 90·24-s + 6·25-s + 30·26-s + 18·27-s + 68·28-s + ⋯
 L(s)  = 1 − 4.24·2-s + 1.73·3-s + 17/2·4-s + 0.894·5-s − 7.34·6-s + 1.51·7-s − 10.6·8-s + 7/3·9-s − 3.79·10-s − 3.01·11-s + 14.7·12-s − 1.38·13-s − 6.41·14-s + 1.54·15-s + 10·16-s − 0.242·17-s − 9.89·18-s − 0.458·19-s + 7.60·20-s + 2.61·21-s + 12.7·22-s − 2.29·23-s − 18.3·24-s + 6/5·25-s + 5.88·26-s + 3.46·27-s + 12.8·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3418801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$3418801$$    =    $$43^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{43} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 3418801,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.0986474$$ $$L(\frac12)$$ $$\approx$$ $$0.0986474$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 43$,$$F_p(T)$$ is a polynomial of degree 8. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad43$C_2$ $$( 1 + 13 T + p T^{2} )^{2}$$
good2$C_2^2$ $$( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
3$D_4\times C_2$ $$1 - p T + 2 T^{2} - p T^{3} + 13 T^{4} - p^{2} T^{5} + 2 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8}$$
5$D_4\times C_2$ $$1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 8 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
7$D_4\times C_2$ $$1 - 4 T + 3 T^{2} + 4 T^{3} + 8 T^{4} + 4 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 25 p T^{5} - 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 + T - 2 T^{2} - 31 T^{3} - 297 T^{4} - 31 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
19$D_4\times C_2$ $$1 + 2 T - 30 T^{2} - 8 T^{3} + 719 T^{4} - 8 p T^{5} - 30 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 11 T + 2 p T^{2} + 319 T^{3} + 2313 T^{4} + 319 p T^{5} + 2 p^{3} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 27 p T^{5} - 56 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
47$D_{4}$ $$( 1 - 9 T + 103 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 - 5 T - 86 T^{2} - 25 T^{3} + 8187 T^{4} - 25 p T^{5} - 86 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 11 p T^{5} - 110 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
67$D_4\times C_2$ $$1 - T - 122 T^{2} + 11 T^{3} + 10573 T^{4} + 11 p T^{5} - 122 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 + 3 T + 16 T^{2} - 447 T^{3} - 5631 T^{4} - 447 p T^{5} + 16 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 27 p T^{5} - 128 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 5 T - 138 T^{2} - 25 T^{3} + 18353 T^{4} - 25 p T^{5} - 138 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
83$D_4\times C_2$ $$1 + 3 T + 52 T^{2} - 627 T^{3} - 5787 T^{4} - 627 p T^{5} + 52 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 + 3 T - 160 T^{2} - 27 T^{3} + 19839 T^{4} - 27 p T^{5} - 160 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}