Properties

 Degree 4 Conductor $2 \cdot 3491$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 2

Origins

Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s + 4-s − 5·5-s + 6·6-s − 4·7-s + 2·8-s + 2·9-s + 10·10-s − 3·11-s − 3·12-s + 8·14-s + 15·15-s − 3·16-s − 2·17-s − 4·18-s − 19-s − 5·20-s + 12·21-s + 6·22-s + 23-s − 6·24-s + 10·25-s + 6·27-s − 4·28-s − 6·29-s − 30·30-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.23·5-s + 2.44·6-s − 1.51·7-s + 0.707·8-s + 2/3·9-s + 3.16·10-s − 0.904·11-s − 0.866·12-s + 2.13·14-s + 3.87·15-s − 3/4·16-s − 0.485·17-s − 0.942·18-s − 0.229·19-s − 1.11·20-s + 2.61·21-s + 1.27·22-s + 0.208·23-s − 1.22·24-s + 2·25-s + 1.15·27-s − 0.755·28-s − 1.11·29-s − 5.47·30-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6982 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 6982 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

Invariants

 $$d$$ = $$4$$ $$N$$ = $$6982$$    =    $$2 \cdot 3491$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{6982} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(4,\ 6982,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3491\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3491\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ $$( 1 + T )( 1 + T + p T^{2} )$$
3491$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 92 T + p T^{2} )$$
good3$D_{4}$ $$1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$C_4$ $$1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$D_{4}$ $$1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
29$D_{4}$ $$1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 2 T - 24 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 13 T + 95 T^{2} + 13 p T^{3} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 2 T - 54 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
79$D_{4}$ $$1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 7 T + 98 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 22 T + 286 T^{2} + 22 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 11 T + 113 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}