# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s − 9-s + 10-s − 2·11-s + 12-s + 4·13-s + 14-s + 15-s − 16-s + 18-s + 3·19-s + 20-s + 21-s + 2·22-s − 4·23-s − 3·24-s + 25-s − 4·26-s + 28-s + 29-s − 30-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.235·18-s + 0.688·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 0.185·29-s − 0.182·30-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$472$$    =    $$2^{3} \cdot 59$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{472} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 472,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.2274474476$ $L(\frac12)$ $\approx$ $0.2274474476$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;59\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + T + p T^{2}$$
59$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 12 T + p T^{2} )$$
good3$D_{4}$ $$1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + T + p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - p T^{2} )^{2}$$
19$D_{4}$ $$1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$D_{4}$ $$1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
41$D_{4}$ $$1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
47$D_{4}$ $$1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
79$D_{4}$ $$1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}