# Properties

 Degree 4 Conductor 53623 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 3

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 4·3-s + 4-s − 4·5-s + 8·6-s − 7·7-s + 7·9-s + 8·10-s − 2·11-s − 4·12-s − 4·13-s + 14·14-s + 16·15-s + 16-s − 4·17-s − 14·18-s − 4·19-s − 4·20-s + 28·21-s + 4·22-s − 10·23-s + 5·25-s + 8·26-s − 4·27-s − 7·28-s − 32·30-s − 8·31-s + ⋯
 L(s)  = 1 − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 3.26·6-s − 2.64·7-s + 7/3·9-s + 2.52·10-s − 0.603·11-s − 1.15·12-s − 1.10·13-s + 3.74·14-s + 4.13·15-s + 1/4·16-s − 0.970·17-s − 3.29·18-s − 0.917·19-s − 0.894·20-s + 6.11·21-s + 0.852·22-s − 2.08·23-s + 25-s + 1.56·26-s − 0.769·27-s − 1.32·28-s − 5.84·30-s − 1.43·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 53623 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 53623 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$53623$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{53623} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 3 Selberg data = $(4,\ 53623,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 53623$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 53623$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad53623$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 48 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + p T + p T^{2} )$$
5$C_2^2$ $$1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
17$D_{4}$ $$1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} )$$
29$C_2^2$ $$1 + 32 T^{2} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
47$D_{4}$ $$1 - T + 70 T^{2} - p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 11 T + 76 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 3 T + 145 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 4 T + 148 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 5 T - 35 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}