# Properties

 Degree $4$ Conductor $7529$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $2$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 4·3-s + 4-s − 2·5-s + 8·6-s − 4·7-s + 8·9-s + 4·10-s − 4·11-s − 4·12-s − 3·13-s + 8·14-s + 8·15-s + 16-s − 4·17-s − 16·18-s − 8·19-s − 2·20-s + 16·21-s + 8·22-s + 2·23-s − 2·25-s + 6·26-s − 12·27-s − 4·28-s + 7·29-s − 16·30-s + ⋯
 L(s)  = 1 − 1.41·2-s − 2.30·3-s + 1/2·4-s − 0.894·5-s + 3.26·6-s − 1.51·7-s + 8/3·9-s + 1.26·10-s − 1.20·11-s − 1.15·12-s − 0.832·13-s + 2.13·14-s + 2.06·15-s + 1/4·16-s − 0.970·17-s − 3.77·18-s − 1.83·19-s − 0.447·20-s + 3.49·21-s + 1.70·22-s + 0.417·23-s − 2/5·25-s + 1.17·26-s − 2.30·27-s − 0.755·28-s + 1.29·29-s − 2.92·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$7529$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{7529} (1, \cdot )$ Sato-Tate group: $\mathrm{USp}(4)$ Primitive: yes Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 7529,\ (\ :1/2, 1/2),\ 1)$$

## 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)$
bad7529$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 15 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$D_{4}$ $$1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
37$D_{4}$ $$1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 8 T + 79 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 9 T + 74 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 8 T + 81 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 12 T + 108 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 5 T + 132 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$