# Properties

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

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 110·5-s − 296·11-s + 4.44e3·19-s + 8.97e3·25-s − 540·29-s + 4.09e3·31-s + 4.79e3·41-s + 8.65e3·49-s + 3.25e4·55-s + 7.94e4·59-s − 8.45e4·61-s − 8.49e3·71-s + 7.05e4·79-s − 1.70e5·89-s − 4.88e5·95-s + 8.59e3·101-s + 7.19e4·109-s − 2.56e5·121-s − 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s + 5.94e4·145-s + 149-s + 151-s − 4.50e5·155-s + ⋯
 L(s)  = 1 − 1.96·5-s − 0.737·11-s + 2.82·19-s + 2.87·25-s − 0.119·29-s + 0.765·31-s + 0.445·41-s + 0.514·49-s + 1.45·55-s + 2.97·59-s − 2.91·61-s − 0.200·71-s + 1.27·79-s − 2.28·89-s − 5.55·95-s + 0.0838·101-s + 0.580·109-s − 1.59·121-s − 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 0.234·145-s + 3.69e−6·149-s + 3.56e−6·151-s − 1.50·155-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$518400$$    =    $$2^{8} \cdot 3^{4} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{720} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 518400,\ (\ :5/2, 5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.262164682$$ $$L(\frac12)$$ $$\approx$$ $$1.262164682$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
5$C_2$ $$1 + 22 p T + p^{5} T^{2}$$
good7$C_2^2$ $$1 - 8650 T^{2} + p^{10} T^{4}$$
11$C_2$ $$( 1 + 148 T + p^{5} T^{2} )^{2}$$
13$C_2^2$ $$1 - 274730 T^{2} + p^{10} T^{4}$$
17$C_2^2$ $$1 + 1354590 T^{2} + p^{10} T^{4}$$
19$C_2$ $$( 1 - 2220 T + p^{5} T^{2} )^{2}$$
23$C_2^2$ $$1 - 11320170 T^{2} + p^{10} T^{4}$$
29$C_2$ $$( 1 + 270 T + p^{5} T^{2} )^{2}$$
31$C_2$ $$( 1 - 2048 T + p^{5} T^{2} )^{2}$$
37$C_2^2$ $$1 - 119573530 T^{2} + p^{10} T^{4}$$
41$C_2$ $$( 1 - 2398 T + p^{5} T^{2} )^{2}$$
43$C_2^2$ $$1 - 288754450 T^{2} + p^{10} T^{4}$$
47$C_2^2$ $$1 - 344584890 T^{2} + p^{10} T^{4}$$
53$C_2^2$ $$1 - 827605690 T^{2} + p^{10} T^{4}$$
59$C_2$ $$( 1 - 39740 T + p^{5} T^{2} )^{2}$$
61$C_2$ $$( 1 + 42298 T + p^{5} T^{2} )^{2}$$
67$C_2^2$ $$1 - 1669968610 T^{2} + p^{10} T^{4}$$
71$C_2$ $$( 1 + 4248 T + p^{5} T^{2} )^{2}$$
73$C_2^2$ $$1 - 3239892370 T^{2} + p^{10} T^{4}$$
79$C_2$ $$( 1 - 35280 T + p^{5} T^{2} )^{2}$$
83$C_2^2$ $$1 - 7103795010 T^{2} + p^{10} T^{4}$$
89$C_2$ $$( 1 + 85210 T + p^{5} T^{2} )^{2}$$
97$C_2^2$ $$1 - 7720618690 T^{2} + p^{10} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}