# Properties

 Degree $4$ Conductor $7549$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $2$

# Origins

## Dirichlet series

 L(s)  = 1 − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 7·7-s − 3·8-s + 9-s + 9·10-s − 6·11-s − 8·12-s + 2·13-s + 21·14-s + 6·15-s + 3·16-s − 2·17-s − 3·18-s − 19-s − 12·20-s + 14·21-s + 18·22-s − 23-s + 6·24-s + 4·25-s − 6·26-s − 2·27-s − 28·28-s + ⋯
 L(s)  = 1 − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 2.64·7-s − 1.06·8-s + 1/3·9-s + 2.84·10-s − 1.80·11-s − 2.30·12-s + 0.554·13-s + 5.61·14-s + 1.54·15-s + 3/4·16-s − 0.485·17-s − 0.707·18-s − 0.229·19-s − 2.68·20-s + 3.05·21-s + 3.83·22-s − 0.208·23-s + 1.22·24-s + 4/5·25-s − 1.17·26-s − 0.384·27-s − 5.29·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7549 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7549 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$7549$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{7549} (1, \cdot )$ Sato-Tate group: $\mathrm{USp}(4)$ Primitive: yes Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 7549,\ (\ :1/2, 1/2),\ 1)$$

## 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)$
bad7549$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 148 T + p T^{2} )$$
good2$C_2^2$ $$1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
3$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + p T + p T^{2} )$$
5$D_{4}$ $$1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
17$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$D_{4}$ $$1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 40 T^{2} + p^{2} T^{4}$$
41$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$D_{4}$ $$1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 10 T + 62 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 23 T + 264 T^{2} - 23 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 9 T + 145 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 13 T + 167 T^{2} - 13 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$