# Properties

 Degree 4 Conductor $3 \cdot 83$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 2·3-s + 4-s + 4·6-s − 7-s + 4·9-s + 11-s − 2·12-s + 2·14-s + 16-s − 17-s − 8·18-s − 6·19-s + 2·21-s − 2·22-s − 2·25-s − 5·27-s − 28-s + 3·29-s + 3·31-s + 2·32-s − 2·33-s + 2·34-s + 4·36-s − 5·37-s + 12·38-s + 4·41-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s − 0.377·7-s + 4/3·9-s + 0.301·11-s − 0.577·12-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.88·18-s − 1.37·19-s + 0.436·21-s − 0.426·22-s − 2/5·25-s − 0.962·27-s − 0.188·28-s + 0.557·29-s + 0.538·31-s + 0.353·32-s − 0.348·33-s + 0.342·34-s + 2/3·36-s − 0.821·37-s + 1.94·38-s + 0.624·41-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$249$$    =    $$3 \cdot 83$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{249} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 249,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.1315495070$ $L(\frac12)$ $\approx$ $0.1315495070$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;83\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;83\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ $$( 1 - T )( 1 + p T + p T^{2} )$$
83$C_1$$\times$$C_2$ $$( 1 - T )( 1 + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$V_4$ $$1 + 2 T^{2} + p^{2} T^{4}$$
7$D_{4}$ $$1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$V_4$ $$1 - 26 T^{2} + p^{2} T^{4}$$
29$V_4$ $$1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
41$D_{4}$ $$1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
53$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
59$D_{4}$ $$1 + 13 T + 106 T^{2} + 13 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 3 T + 48 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
71$D_{4}$ $$1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$D_{4}$ $$1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
97$D_{4}$ $$1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}