# Properties

 Degree 4 Conductor 997 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s − 2·5-s + 6·6-s − 2·7-s + 4·8-s + 4·9-s + 4·10-s + 4·11-s − 8·13-s + 4·14-s + 6·15-s − 4·16-s − 8·18-s + 19-s + 6·21-s − 8·22-s − 2·23-s − 12·24-s + 16·26-s − 6·27-s − 29-s − 12·30-s − 3·31-s − 12·33-s + 4·35-s − 3·37-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s − 0.894·5-s + 2.44·6-s − 0.755·7-s + 1.41·8-s + 4/3·9-s + 1.26·10-s + 1.20·11-s − 2.21·13-s + 1.06·14-s + 1.54·15-s − 16-s − 1.88·18-s + 0.229·19-s + 1.30·21-s − 1.70·22-s − 0.417·23-s − 2.44·24-s + 3.13·26-s − 1.15·27-s − 0.185·29-s − 2.19·30-s − 0.538·31-s − 2.08·33-s + 0.676·35-s − 0.493·37-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$997$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{997} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 997,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 997$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 997$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad997$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 28 T + p T^{2} )$$
good2$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_4$ $$1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
17$V_4$ $$1 - 14 T^{2} + p^{2} T^{4}$$
19$D_{4}$ $$1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
37$D_{4}$ $$1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 11 T + 88 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
59$D_{4}$ $$1 - 5 T + 85 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$D_{4}$ $$1 - 11 T + 110 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 13 T + 82 T^{2} + 13 p T^{3} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
79$D_{4}$ $$1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 9 T + 178 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}