# Properties

 Degree 8 Conductor $3^{4} \cdot 7^{8} \cdot 41^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 4·3-s − 2·4-s + 3·5-s + 4·6-s − 2·8-s + 10·9-s + 3·10-s − 5·11-s − 8·12-s + 5·13-s + 12·15-s − 16-s − 5·17-s + 10·18-s + 6·19-s − 6·20-s − 5·22-s + 3·23-s − 8·24-s − 8·25-s + 5·26-s + 20·27-s + 2·29-s + 12·30-s + 11·31-s − 3·32-s + ⋯
 L(s)  = 1 + 0.707·2-s + 2.30·3-s − 4-s + 1.34·5-s + 1.63·6-s − 0.707·8-s + 10/3·9-s + 0.948·10-s − 1.50·11-s − 2.30·12-s + 1.38·13-s + 3.09·15-s − 1/4·16-s − 1.21·17-s + 2.35·18-s + 1.37·19-s − 1.34·20-s − 1.06·22-s + 0.625·23-s − 1.63·24-s − 8/5·25-s + 0.980·26-s + 3.84·27-s + 0.371·29-s + 2.19·30-s + 1.97·31-s − 0.530·32-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 41^{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 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$3^{4} \cdot 7^{8} \cdot 41^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{6027} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 3^{4} \cdot 7^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $\approx$ $28.28457818$ $L(\frac12)$ $\approx$ $28.28457818$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;7,\;41\}$, $$F_p$$ is a polynomial of degree 8. If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ $$( 1 - T )^{4}$$
7 $$1$$
41$C_1$ $$( 1 - T )^{4}$$
good2$C_2 \wr S_4$ $$1 - T + 3 T^{2} - 3 T^{3} + p^{3} T^{4} - 3 p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
5$C_2 \wr S_4$ $$1 - 3 T + 17 T^{2} - 38 T^{3} + 122 T^{4} - 38 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 + 5 T + 42 T^{2} + 145 T^{3} + 690 T^{4} + 145 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 - 5 T + 3 p T^{2} - 148 T^{3} + 764 T^{4} - 148 p T^{5} + 3 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 + 5 T + 58 T^{2} + 247 T^{3} + 1398 T^{4} + 247 p T^{5} + 58 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 6 T + 72 T^{2} - 330 T^{3} + 2022 T^{4} - 330 p T^{5} + 72 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 - 3 T + 47 T^{2} - 262 T^{3} + 1088 T^{4} - 262 p T^{5} + 47 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 - 2 T + 54 T^{2} - 130 T^{3} + 1941 T^{4} - 130 p T^{5} + 54 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 - 11 T + 150 T^{2} - 967 T^{3} + 7190 T^{4} - 967 p T^{5} + 150 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 - 4 T + 110 T^{2} - 508 T^{3} + 5379 T^{4} - 508 p T^{5} + 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 + 19 T + 230 T^{2} + 1807 T^{3} + 12846 T^{4} + 1807 p T^{5} + 230 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 - 4 T + 38 T^{2} - 156 T^{3} - 1117 T^{4} - 156 p T^{5} + 38 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 - 9 T + 177 T^{2} - 1048 T^{3} + 12638 T^{4} - 1048 p T^{5} + 177 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 - 16 T + 288 T^{2} - 2884 T^{3} + 27038 T^{4} - 2884 p T^{5} + 288 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 - 27 T + 440 T^{2} - 4789 T^{3} + 42114 T^{4} - 4789 p T^{5} + 440 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 + 13 T + 261 T^{2} + 2600 T^{3} + 25998 T^{4} + 2600 p T^{5} + 261 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - T + 108 T^{2} - 729 T^{3} + 5138 T^{4} - 729 p T^{5} + 108 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 - 25 T + 490 T^{2} - 6051 T^{3} + 61430 T^{4} - 6051 p T^{5} + 490 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 + T + 113 T^{2} + 424 T^{3} + 9198 T^{4} + 424 p T^{5} + 113 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 + 268 T^{2} + 144 T^{3} + 30726 T^{4} + 144 p T^{5} + 268 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 10 T + 316 T^{2} - 2606 T^{3} + 40470 T^{4} - 2606 p T^{5} + 316 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 + 15 T + 319 T^{2} + 3494 T^{3} + 45974 T^{4} + 3494 p T^{5} + 319 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}