# Properties

 Degree 4 Conductor 5641 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 + 2·4-s − 5·5-s + 8·6-s − 4·7-s − 4·8-s + 8·9-s + 10·10-s − 11-s − 8·12-s + 8·14-s + 20·15-s + 8·16-s + 4·17-s − 16·18-s − 3·19-s − 10·20-s + 16·21-s + 2·22-s − 9·23-s + 16·24-s + 10·25-s − 12·27-s − 8·28-s − 10·29-s − 40·30-s + ⋯
 L(s)  = 1 − 1.41·2-s − 2.30·3-s + 4-s − 2.23·5-s + 3.26·6-s − 1.51·7-s − 1.41·8-s + 8/3·9-s + 3.16·10-s − 0.301·11-s − 2.30·12-s + 2.13·14-s + 5.16·15-s + 2·16-s + 0.970·17-s − 3.77·18-s − 0.688·19-s − 2.23·20-s + 3.49·21-s + 0.426·22-s − 1.87·23-s + 3.26·24-s + 2·25-s − 2.30·27-s − 1.51·28-s − 1.85·29-s − 7.30·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$5641$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{5641} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(4,\ 5641,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5641$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 5641$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5641$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 70 T + p T^{2} )$$
good2$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$C_2^2$ $$1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
5$C_4$ $$1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7$C_2$ $$( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} )$$
11$D_{4}$ $$1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 5 T^{2} + p^{2} T^{4}$$
17$D_{4}$ $$1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 28 T^{2} + p^{2} T^{4}$$
61$D_{4}$ $$1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 20 T + 244 T^{2} + 20 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 3 T - 101 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}