# Properties

 Degree 4 Conductor $2^{2} \cdot 3 \cdot 59$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·3-s − 4-s + 5-s − 3·7-s + 2·11-s + 2·12-s − 2·15-s + 16-s + 4·17-s − 5·19-s − 20-s + 6·21-s + 8·23-s + 25-s + 5·27-s + 3·28-s − 5·29-s − 6·31-s − 4·33-s − 3·35-s + 2·37-s − 3·41-s + 2·43-s − 2·44-s − 2·47-s − 2·48-s + 3·49-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1/2·4-s + 0.447·5-s − 1.13·7-s + 0.603·11-s + 0.577·12-s − 0.516·15-s + 1/4·16-s + 0.970·17-s − 1.14·19-s − 0.223·20-s + 1.30·21-s + 1.66·23-s + 1/5·25-s + 0.962·27-s + 0.566·28-s − 0.928·29-s − 1.07·31-s − 0.696·33-s − 0.507·35-s + 0.328·37-s − 0.468·41-s + 0.304·43-s − 0.301·44-s − 0.291·47-s − 0.288·48-s + 3/7·49-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 708 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 708 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$708$$    =    $$2^{2} \cdot 3 \cdot 59$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{708} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 708,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.3253436263$ $L(\frac12)$ $\approx$ $0.3253436263$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;59\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + T^{2}$$
3$C_1$$\times$$C_2$ $$( 1 + T )( 1 + T + p T^{2} )$$
59$C_1$$\times$$C_2$ $$( 1 + T )( 1 + p T^{2} )$$
good5$D_{4}$ $$1 - T - p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$D_{4}$ $$1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$D_{4}$ $$1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
53$D_{4}$ $$1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
71$D_{4}$ $$1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 13 T + p T^{2} )$$
83$D_{4}$ $$1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}