# Properties

 Degree $4$ Conductor $2457$ Sign $-1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3-s + 4-s − 2·5-s + 2·6-s − 3·7-s + 9-s + 4·10-s − 12-s − 7·13-s + 6·14-s + 2·15-s + 16-s − 2·18-s − 2·20-s + 3·21-s − 2·23-s + 2·25-s + 14·26-s − 27-s − 3·28-s + 2·29-s − 4·30-s + 4·31-s + 2·32-s + 6·35-s + 36-s + ⋯
 L(s)  = 1 − 1.41·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 1/3·9-s + 1.26·10-s − 0.288·12-s − 1.94·13-s + 1.60·14-s + 0.516·15-s + 1/4·16-s − 0.471·18-s − 0.447·20-s + 0.654·21-s − 0.417·23-s + 2/5·25-s + 2.74·26-s − 0.192·27-s − 0.566·28-s + 0.371·29-s − 0.730·30-s + 0.718·31-s + 0.353·32-s + 1.01·35-s + 1/6·36-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2457$$    =    $$3^{3} \cdot 7 \cdot 13$$ Sign: $-1$ Motivic weight: $$1$$ Character: $\chi_{2457} (1, \cdot )$ Sato-Tate group: $\mathrm{USp}(4)$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 2457,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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)$
bad3$C_1$ $$1 + T$$
7$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 4 T + p T^{2} )$$
13$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 6 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$D_{4}$ $$1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
31$D_{4}$ $$1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
43$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 2 T + 38 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 2 T - 46 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
61$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$D_{4}$ $$1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$D_{4}$ $$1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$