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 + 8·17-s + 4·19-s − 2·23-s − 4·25-s + 8·29-s − 4·31-s − 8·35-s − 4·37-s + 2·41-s + 8·43-s − 12·47-s + 10·49-s + 16·53-s − 4·55-s − 12·59-s − 8·61-s − 8·67-s + 18·71-s + 8·73-s + 8·77-s + 16·85-s + 18·89-s + 8·95-s + 8·97-s + ⋯
 L(s)  = 1 + 0.894·5-s − 1.51·7-s − 0.603·11-s + 1.94·17-s + 0.917·19-s − 0.417·23-s − 4/5·25-s + 1.48·29-s − 0.718·31-s − 1.35·35-s − 0.657·37-s + 0.312·41-s + 1.21·43-s − 1.75·47-s + 10/7·49-s + 2.19·53-s − 0.539·55-s − 1.56·59-s − 1.02·61-s − 0.977·67-s + 2.13·71-s + 0.936·73-s + 0.911·77-s + 1.73·85-s + 1.90·89-s + 0.820·95-s + 0.812·97-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 = $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $\approx$ $6.175981783$ $L(\frac12)$ $\approx$ $6.175981783$ $L(\frac{3}{2})$ 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$C_2 \wr S_4$ $$1 - 2 T + 8 T^{2} - 24 T^{3} + 33 T^{4} - 24 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 + 2 T + 20 T^{2} - 20 T^{3} + 125 T^{4} - 20 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 + 16 T^{2} - 32 T^{3} + 126 T^{4} - 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 - 8 T + 44 T^{2} - 184 T^{3} + 854 T^{4} - 184 p T^{5} + 44 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr S_4$ $$1 - 4 T + 34 T^{2} - 72 T^{3} + 699 T^{4} - 72 p T^{5} + 34 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 + 2 T + 8 T^{2} + 4 T^{3} + 761 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 - 8 T + 80 T^{2} - 296 T^{3} + 2206 T^{4} - 296 p T^{5} + 80 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 + 4 T + 70 T^{2} + 448 T^{3} + 2407 T^{4} + 448 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 + 4 T + 118 T^{2} + 344 T^{3} + 5975 T^{4} + 344 p T^{5} + 118 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 - 2 T + 92 T^{2} - 80 T^{3} + 4093 T^{4} - 80 p T^{5} + 92 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 - 8 T + 160 T^{2} - 952 T^{3} + 10046 T^{4} - 952 p T^{5} + 160 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 + 12 T + 200 T^{2} + 1484 T^{3} + 13950 T^{4} + 1484 p T^{5} + 200 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
59$C_2 \wr S_4$ $$1 + 12 T + 248 T^{2} + 1916 T^{3} + 21870 T^{4} + 1916 p T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 + 8 T + 100 T^{2} + 440 T^{3} + 3478 T^{4} + 440 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 + 8 T + 88 T^{2} + 8 T^{3} + 814 T^{4} + 8 p T^{5} + 88 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - 18 T + 296 T^{2} - 2724 T^{3} + 27369 T^{4} - 2724 p T^{5} + 296 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 - 8 T + 136 T^{2} - 1032 T^{3} + 11166 T^{4} - 1032 p T^{5} + 136 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 + 160 T^{2} - 848 T^{3} + 11886 T^{4} - 848 p T^{5} + 160 p^{2} T^{6} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 - 40 T^{2} + 16 T^{3} + 13134 T^{4} + 16 p T^{5} - 40 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 18 T + 404 T^{2} - 4416 T^{3} + 54549 T^{4} - 4416 p T^{5} + 404 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 - 8 T + 244 T^{2} - 2200 T^{3} + 29798 T^{4} - 2200 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}