# Properties

 Degree 4 Conductor 691 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s − 4·5-s − 7-s − 3·8-s + 2·9-s + 4·10-s + 2·11-s + 2·13-s + 14-s + 16-s − 2·18-s + 2·19-s − 4·20-s − 2·22-s + 2·25-s − 2·26-s − 28-s + 3·29-s − 2·31-s + 32-s + 4·35-s + 2·36-s − 3·37-s − 2·38-s + 12·40-s + 8·41-s + ⋯
 L(s)  = 1 − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 1.06·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s + 0.458·19-s − 0.894·20-s − 0.426·22-s + 2/5·25-s − 0.392·26-s − 0.188·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s + 0.676·35-s + 1/3·36-s − 0.493·37-s − 0.324·38-s + 1.89·40-s + 1.24·41-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 691 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 691 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$691$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{691} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 691,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.2939463892$ $L(\frac12)$ $\approx$ $0.2939463892$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 691$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 691$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad691$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 28 T + p T^{2} )$$
good2$D_{4}$ $$1 + T + p T^{3} + p^{2} T^{4}$$
3$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
7$D_{4}$ $$1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$V_4$ $$1 - 10 T^{2} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
43$D_{4}$ $$1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
47$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 10 T + 42 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 5 T + 96 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 7 T + 118 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}