Properties

 Degree 8 Conductor $2^{20} \cdot 3^{12} \cdot 7^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 − 2·5-s − 4·7-s + 2·11-s − 4·13-s − 4·17-s + 4·19-s + 10·23-s − 4·25-s + 4·29-s − 8·31-s + 8·35-s − 8·37-s − 2·41-s − 4·43-s + 8·47-s + 10·49-s − 16·53-s − 4·55-s + 24·59-s − 8·61-s + 8·65-s + 4·67-s + 10·71-s + 4·73-s − 8·77-s + 4·79-s + 20·83-s + ⋯
 L(s)  = 1 − 0.894·5-s − 1.51·7-s + 0.603·11-s − 1.10·13-s − 0.970·17-s + 0.917·19-s + 2.08·23-s − 4/5·25-s + 0.742·29-s − 1.43·31-s + 1.35·35-s − 1.31·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 10/7·49-s − 2.19·53-s − 0.539·55-s + 3.12·59-s − 1.02·61-s + 0.992·65-s + 0.488·67-s + 1.18·71-s + 0.468·73-s − 0.911·77-s + 0.450·79-s + 2.19·83-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{20} \cdot 3^{12} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{6048} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $\approx$ $1.509662397$ $L(\frac12)$ $\approx$ $1.509662397$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ 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$C_1$ $$( 1 + T )^{4}$$
good5$C_2 \wr S_4$ $$1 + 2 T + 8 T^{2} + 4 p T^{3} + 57 T^{4} + 4 p^{2} T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 - 2 T + 12 T^{2} + 48 T^{3} - 51 T^{4} + 48 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 + 4 T + 12 T^{2} - 20 T^{3} - 58 T^{4} - 20 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 + 4 T + 36 T^{2} + 124 T^{3} + 694 T^{4} + 124 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 4 T + 26 T^{2} + 40 T^{3} + 3 T^{4} + 40 p T^{5} + 26 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 - 10 T + 96 T^{2} - 576 T^{3} + 3385 T^{4} - 576 p T^{5} + 96 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 - 4 T + 76 T^{2} - 172 T^{3} + 2694 T^{4} - 172 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 + 8 T + 94 T^{2} + 392 T^{3} + 3303 T^{4} + 392 p T^{5} + 94 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 + 8 T + 62 T^{2} - 40 T^{3} - 121 T^{4} - 40 p T^{5} + 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 + 2 T + 108 T^{2} + 4 T^{3} + 5237 T^{4} + 4 p T^{5} + 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 + 4 T + 68 T^{2} - 220 T^{3} + 1014 T^{4} - 220 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 - 8 T + 124 T^{2} - 1016 T^{3} + 7926 T^{4} - 1016 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 + 16 T + 148 T^{2} + 1136 T^{3} + 8534 T^{4} + 1136 p T^{5} + 148 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 - 24 T + 348 T^{2} - 3464 T^{3} + 29158 T^{4} - 3464 p T^{5} + 348 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 + 8 T + 116 T^{2} + 824 T^{3} + 7478 T^{4} + 824 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 - 4 T + 100 T^{2} - 692 T^{3} + 4454 T^{4} - 692 p T^{5} + 100 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - 10 T + 192 T^{2} - 1424 T^{3} + 19193 T^{4} - 1424 p T^{5} + 192 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 - 4 T + 44 T^{2} - 268 T^{3} + 10310 T^{4} - 268 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 - 4 T + 20 T^{2} - 404 T^{3} + 11814 T^{4} - 404 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 - 20 T + 436 T^{2} - 4932 T^{3} + 57734 T^{4} - 4932 p T^{5} + 436 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 + 26 T + 548 T^{2} + 7140 T^{3} + 80541 T^{4} + 7140 p T^{5} + 548 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 - 8 T + 196 T^{2} - 1688 T^{3} + 26118 T^{4} - 1688 p T^{5} + 196 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}