# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 14·4-s − 18·9-s − 32·11-s + 51·16-s + 192·19-s − 376·29-s − 240·31-s − 252·36-s + 200·41-s − 448·44-s − 98·49-s − 208·59-s − 936·61-s − 420·64-s − 272·71-s + 2.68e3·76-s + 864·79-s + 243·81-s + 2.80e3·89-s + 576·99-s − 328·101-s − 280·109-s − 5.26e3·116-s + 1.71e3·121-s − 3.36e3·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 + 7/4·4-s − 2/3·9-s − 0.877·11-s + 0.796·16-s + 2.31·19-s − 2.40·29-s − 1.39·31-s − 7/6·36-s + 0.761·41-s − 1.53·44-s − 2/7·49-s − 0.458·59-s − 1.96·61-s − 0.820·64-s − 0.454·71-s + 4.05·76-s + 1.23·79-s + 1/3·81-s + 3.34·89-s + 0.584·99-s − 0.323·101-s − 0.246·109-s − 4.21·116-s + 1.28·121-s − 2.43·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$3^{4} \cdot 5^{8} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{525} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$ $$L(2)$$ $$\approx$$ $$0.3598166946$$ $$L(\frac12)$$ $$\approx$$ $$0.3598166946$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
5 $$1$$
7$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
good2$D_4\times C_2$ $$1 - 7 p T^{2} + 145 T^{4} - 7 p^{7} T^{6} + p^{12} T^{8}$$
11$D_{4}$ $$( 1 + 16 T - 474 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 5836 T^{2} + 17983510 T^{4} - 5836 p^{6} T^{6} + p^{12} T^{8}$$
17$D_4\times C_2$ $$1 - 700 p T^{2} + 83185606 T^{4} - 700 p^{7} T^{6} + p^{12} T^{8}$$
19$D_{4}$ $$( 1 - 96 T + 13974 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 30044 T^{2} + 519364774 T^{4} - 30044 p^{6} T^{6} + p^{12} T^{8}$$
29$D_{4}$ $$( 1 + 188 T + 34286 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$D_{4}$ $$( 1 + 120 T + 60590 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 124204 T^{2} + 8380919670 T^{4} - 124204 p^{6} T^{6} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 - 100 T + 89142 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 112876 T^{2} + 6993047350 T^{4} - 112876 p^{6} T^{6} + p^{12} T^{8}$$
47$D_4\times C_2$ $$1 + 43524 T^{2} + 9878986694 T^{4} + 43524 p^{6} T^{6} + p^{12} T^{8}$$
53$D_4\times C_2$ $$1 - 139244 T^{2} + 23569194934 T^{4} - 139244 p^{6} T^{6} + p^{12} T^{8}$$
59$D_{4}$ $$( 1 + 104 T + 80534 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
61$D_{4}$ $$( 1 + 468 T + 494606 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 + 244724 T^{2} + 162973601494 T^{4} + 244724 p^{6} T^{6} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 136 T + 540446 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 231908 T^{2} + 102056920486 T^{4} + 231908 p^{6} T^{6} + p^{12} T^{8}$$
79$D_{4}$ $$( 1 - 432 T + 602142 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 2003724 T^{2} + 1638356284694 T^{4} - 2003724 p^{6} T^{6} + p^{12} T^{8}$$
89$D_{4}$ $$( 1 - 1404 T + 1802390 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 30012 p T^{2} + 3760771737350 T^{4} - 30012 p^{7} T^{6} + p^{12} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}