# Properties

 Degree $4$ Conductor $518400$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·19-s − 5·25-s − 16·31-s + 14·49-s + 4·61-s + 32·79-s + 28·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 + 1.83·19-s − 25-s − 2.87·31-s + 2·49-s + 0.512·61-s + 3.60·79-s + 2.68·109-s − 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$4$$ Conductor: $$518400$$    =    $$2^{8} \cdot 3^{4} \cdot 5^{2}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{720} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 518400,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.76091$$ $$L(\frac12)$$ $$\approx$$ $$1.76091$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
5$C_2$ $$1 + p T^{2}$$
good7$C_2$ $$( 1 - p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - p T^{2} )^{2}$$
17$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
23$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - p T^{2} )^{2}$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 - p T^{2} )^{2}$$
47$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 86 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - p T^{2} )^{2}$$
79$C_2$ $$( 1 - 16 T + p T^{2} )^{2}$$
83$C_2^2$ $$1 + 154 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 + p T^{2} )^{2}$$
97$C_2$ $$( 1 - p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$