# Properties

 Degree 4 Conductor $5 \cdot 1433$ 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 − 3·5-s + 8·6-s − 7-s + 6·9-s + 6·10-s − 6·11-s − 4·12-s − 3·13-s + 2·14-s + 12·15-s + 16-s − 4·17-s − 12·18-s − 19-s − 3·20-s + 4·21-s + 12·22-s − 7·23-s + 8·25-s + 6·26-s + 4·27-s − 28-s + 6·29-s − 24·30-s + ⋯
 L(s)  = 1 − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.34·5-s + 3.26·6-s − 0.377·7-s + 2·9-s + 1.89·10-s − 1.80·11-s − 1.15·12-s − 0.832·13-s + 0.534·14-s + 3.09·15-s + 1/4·16-s − 0.970·17-s − 2.82·18-s − 0.229·19-s − 0.670·20-s + 0.872·21-s + 2.55·22-s − 1.45·23-s + 8/5·25-s + 1.17·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s − 4.38·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 7165 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 7165 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$7165$$    =    $$5 \cdot 1433$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{7165} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(4,\ 7165,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;1433\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;1433\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 4 T + p T^{2} )$$
1433$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 34 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
7$D_{4}$ $$1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4}$$
11$C_4$ $$1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 3 T + 3 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$D_{4}$ $$1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$D_{4}$ $$1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
61$D_{4}$ $$1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 8 T + 2 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 5 T + 92 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 9 T + 26 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
97$D_{4}$ $$1 + 15 T + 240 T^{2} + 15 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}