# Properties

 Degree 4 Conductor $2^{2} \cdot 2297$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 2

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 4·3-s − 5·5-s + 4·6-s − 4·7-s + 8-s + 7·9-s + 5·10-s − 4·11-s − 13-s + 4·14-s + 20·15-s − 16-s + 3·17-s − 7·18-s − 6·19-s + 16·21-s + 4·22-s + 2·23-s − 4·24-s + 12·25-s + 26-s − 4·27-s − 2·29-s − 20·30-s − 8·31-s + 16·33-s + ⋯
 L(s)  = 1 − 0.707·2-s − 2.30·3-s − 2.23·5-s + 1.63·6-s − 1.51·7-s + 0.353·8-s + 7/3·9-s + 1.58·10-s − 1.20·11-s − 0.277·13-s + 1.06·14-s + 5.16·15-s − 1/4·16-s + 0.727·17-s − 1.64·18-s − 1.37·19-s + 3.49·21-s + 0.852·22-s + 0.417·23-s − 0.816·24-s + 12/5·25-s + 0.196·26-s − 0.769·27-s − 0.371·29-s − 3.65·30-s − 1.43·31-s + 2.78·33-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 9188 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 9188 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$9188$$    =    $$2^{2} \cdot 2297$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{9188} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(4,\ 9188,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;2297\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;2297\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + T + T^{2}$$
2297$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 54 T + p T^{2} )$$
good3$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + p T + p T^{2} )$$
5$D_{4}$ $$1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$D_{4}$ $$1 - 6 T + 15 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 10 T + 112 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
67$D_{4}$ $$1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$D_{4}$ $$1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} )$$
83$D_{4}$ $$1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 9 T + 29 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}