# Properties

 Degree 4 Conductor $2^{3} \cdot 1907$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 2

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 4·3-s + 4-s − 4·5-s + 4·6-s − 3·7-s − 8-s + 7·9-s + 4·10-s − 4·11-s − 4·12-s − 4·13-s + 3·14-s + 16·15-s + 16-s − 4·17-s − 7·18-s − 4·20-s + 12·21-s + 4·22-s − 6·23-s + 4·24-s + 7·25-s + 4·26-s − 4·27-s − 3·28-s + 3·29-s + ⋯
 L(s)  = 1 − 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 1.63·6-s − 1.13·7-s − 0.353·8-s + 7/3·9-s + 1.26·10-s − 1.20·11-s − 1.15·12-s − 1.10·13-s + 0.801·14-s + 4.13·15-s + 1/4·16-s − 0.970·17-s − 1.64·18-s − 0.894·20-s + 2.61·21-s + 0.852·22-s − 1.25·23-s + 0.816·24-s + 7/5·25-s + 0.784·26-s − 0.769·27-s − 0.566·28-s + 0.557·29-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 15256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 15256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$15256$$    =    $$2^{3} \cdot 1907$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{15256} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(4,\ 15256,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;1907\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;1907\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$1 + T$$
1907$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 29 T + p T^{2} )$$
good3$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + p T + p T^{2} )$$
5$D_{4}$ $$1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + T - 65 T^{2} + p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 4 T + 52 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 9 T + 123 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 65 T^{2} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 23 T + 3 p T^{2} - 23 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}