Properties

 Degree 8 Conductor $2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

Origins of factors

Dirichlet series

 L(s)  = 1 − 8·13-s − 16-s + 12·17-s + 8·23-s − 24·29-s − 24·31-s + 4·37-s + 20·43-s + 20·47-s − 8·53-s − 32·59-s + 32·61-s − 4·67-s − 24·73-s − 16·83-s − 8·89-s − 16·97-s + 8·103-s + 8·107-s + 16·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 2.21·13-s − 1/4·16-s + 2.91·17-s + 1.66·23-s − 4.45·29-s − 4.31·31-s + 0.657·37-s + 3.04·43-s + 2.91·47-s − 1.09·53-s − 4.16·59-s + 4.09·61-s − 0.488·67-s − 2.80·73-s − 1.75·83-s − 0.847·89-s − 1.62·97-s + 0.788·103-s + 0.773·107-s + 1.50·113-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \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^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{3150} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$ $L(1)$ $\approx$ $2.647729106$ $L(\frac12)$ $\approx$ $2.647729106$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ $$1 + T^{4}$$
3 $$1$$
5 $$1$$
7$C_2^2$ $$1 + T^{4}$$
good11$D_4\times C_2$ $$1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 - 12 T + 72 T^{2} - 372 T^{3} + 1726 T^{4} - 372 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 + 12 T^{2} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 8 T + 32 T^{2} - 120 T^{3} + 386 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 4 T + 8 T^{2} + 244 T^{3} - 2162 T^{4} + 244 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 20 T + 200 T^{2} - 1780 T^{3} + 13726 T^{4} - 1780 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 20 T + 200 T^{2} - 1860 T^{3} + 15182 T^{4} - 1860 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 + 8 T + 32 T^{2} + 360 T^{3} + 3986 T^{4} + 360 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
61$D_{4}$ $$( 1 - 16 T + 184 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 + 4 T + 8 T^{2} + 260 T^{3} + 8446 T^{4} + 260 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 120 T^{2} + 9074 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 - 52 T^{2} + 11110 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}$$
83$D_4\times C_2$ $$1 + 16 T + 128 T^{2} + 816 T^{3} + 4178 T^{4} + 816 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 + 16 T + 128 T^{2} + 1488 T^{3} + 17282 T^{4} + 1488 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}