Properties

 Degree 4 Conductor $5 \cdot 277$ 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 + 2·4-s − 2·5-s + 6·6-s − 7-s − 4·8-s + 2·9-s + 4·10-s − 6·12-s − 13-s + 2·14-s + 6·15-s + 8·16-s − 4·18-s + 19-s − 4·20-s + 3·21-s + 23-s + 12·24-s − 2·25-s + 2·26-s + 6·27-s − 2·28-s − 5·29-s − 12·30-s − 8·32-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.73·12-s − 0.277·13-s + 0.534·14-s + 1.54·15-s + 2·16-s − 0.942·18-s + 0.229·19-s − 0.894·20-s + 0.654·21-s + 0.208·23-s + 2.44·24-s − 2/5·25-s + 0.392·26-s + 1.15·27-s − 0.377·28-s − 0.928·29-s − 2.19·30-s − 1.41·32-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1385 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1385 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$4$$ $$N$$ = $$1385$$    =    $$5 \cdot 277$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{1385} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 1385,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{5,\;277\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;277\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$$\times$$C_2$ $$( 1 + T )( 1 + T + p T^{2} )$$
277$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 24 T + p T^{2} )$$
good2$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
13$D_{4}$ $$1 + T - 3 T^{2} + p T^{3} + p^{2} T^{4}$$
17$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
19$D_{4}$ $$1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 46 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 - 12 T^{2} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 + 30 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 46 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
71$D_{4}$ $$1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 11 T + 152 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
97$D_{4}$ $$1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}