# Properties

 Degree 8 Conductor $2^{8} \cdot 41^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 4·5-s + 2·9-s + 4·11-s + 8·15-s − 4·17-s + 6·19-s − 12·23-s + 4·25-s − 4·27-s − 4·29-s − 8·31-s + 8·33-s + 16·37-s − 4·41-s + 4·43-s + 8·45-s − 6·47-s − 6·49-s − 8·51-s − 16·53-s + 16·55-s + 12·57-s + 12·59-s + 24·61-s + 28·67-s − 24·69-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1.78·5-s + 2/3·9-s + 1.20·11-s + 2.06·15-s − 0.970·17-s + 1.37·19-s − 2.50·23-s + 4/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s + 1.39·33-s + 2.63·37-s − 0.624·41-s + 0.609·43-s + 1.19·45-s − 0.875·47-s − 6/7·49-s − 1.12·51-s − 2.19·53-s + 2.15·55-s + 1.58·57-s + 1.56·59-s + 3.07·61-s + 3.42·67-s − 2.88·69-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{8} \cdot 41^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{164} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 2^{8} \cdot 41^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $\approx$ $2.84883$ $L(\frac12)$ $\approx$ $2.84883$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;41\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
41$C_1$ $$( 1 + T )^{4}$$
good3$C_2 \wr S_4$ $$1 - 2 T + 2 T^{2} + 4 T^{3} - 8 T^{4} + 4 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
5$C_2 \wr S_4$ $$1 - 4 T + 12 T^{2} - 16 T^{3} + 34 T^{4} - 16 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
7$C_2 \wr S_4$ $$1 + 6 T^{2} + 26 T^{3} + 24 T^{4} + 26 p T^{5} + 6 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 - 4 T + 26 T^{2} - 114 T^{3} + 384 T^{4} - 114 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 + 12 T^{2} - 48 T^{3} + 118 T^{4} - 48 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 + 4 T + 20 T^{2} + 124 T^{3} + 534 T^{4} + 124 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 6 T + 62 T^{2} - 208 T^{3} + 1448 T^{4} - 208 p T^{5} + 62 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 + 12 T + 108 T^{2} + 700 T^{3} + 3718 T^{4} + 700 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 + 4 T + 76 T^{2} + 12 p T^{3} + 2870 T^{4} + 12 p^{2} T^{5} + 76 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 + 8 T + 92 T^{2} + 712 T^{3} + 3846 T^{4} + 712 p T^{5} + 92 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 - 16 T + 212 T^{2} - 1740 T^{3} + 12626 T^{4} - 1740 p T^{5} + 212 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 - 4 T + 124 T^{2} - 244 T^{3} + 6678 T^{4} - 244 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 + 6 T + 126 T^{2} + 640 T^{3} + 8608 T^{4} + 640 p T^{5} + 126 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 + 16 T + 4 p T^{2} + 1824 T^{3} + 15558 T^{4} + 1824 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 - 12 T + 252 T^{2} - 1996 T^{3} + 22582 T^{4} - 1996 p T^{5} + 252 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 - 24 T + 420 T^{2} - 4824 T^{3} + 44086 T^{4} - 4824 p T^{5} + 420 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 - 28 T + 538 T^{2} - 6638 T^{3} + 64208 T^{4} - 6638 p T^{5} + 538 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 + 2 T + 98 T^{2} - 268 T^{3} + 48 p T^{4} - 268 p T^{5} + 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 - 8 T + 212 T^{2} - 1060 T^{3} + 19890 T^{4} - 1060 p T^{5} + 212 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 + 18 T + 366 T^{2} + 4224 T^{3} + 45328 T^{4} + 4224 p T^{5} + 366 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 + 12 T + 252 T^{2} + 1644 T^{3} + 24598 T^{4} + 1644 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 4 T + 228 T^{2} - 796 T^{3} + 326 p T^{4} - 796 p T^{5} + 228 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 - 16 T + 268 T^{2} - 3376 T^{3} + 38118 T^{4} - 3376 p T^{5} + 268 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}