Properties

 Degree 4 Conductor 66161 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 3

Origins

Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s − 4·5-s + 6·6-s − 7·7-s + 4·8-s + 3·9-s + 8·10-s − 4·11-s − 5·13-s + 14·14-s + 12·15-s − 4·16-s − 17-s − 6·18-s − 8·19-s + 21·21-s + 8·22-s + 3·23-s − 12·24-s + 6·25-s + 10·26-s − 4·29-s − 24·30-s − 12·31-s + 12·33-s + 2·34-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s − 1.78·5-s + 2.44·6-s − 2.64·7-s + 1.41·8-s + 9-s + 2.52·10-s − 1.20·11-s − 1.38·13-s + 3.74·14-s + 3.09·15-s − 16-s − 0.242·17-s − 1.41·18-s − 1.83·19-s + 4.58·21-s + 1.70·22-s + 0.625·23-s − 2.44·24-s + 6/5·25-s + 1.96·26-s − 0.742·29-s − 4.38·30-s − 2.15·31-s + 2.08·33-s + 0.342·34-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 66161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 66161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$4$$ $$N$$ = $$66161$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{66161} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 3 Selberg data = $(4,\ 66161,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 66161$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 66161$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad66161$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 478 T + p T^{2} )$$
good2$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_2$ $$( 1 + p T^{2} )( 1 + p T + p T^{2} )$$
5$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$D_{4}$ $$1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$D_{4}$ $$1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + T - 20 T^{2} + p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 5 T + 48 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
41$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
43$D_{4}$ $$1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4}$$
61$C_2^2$ $$1 - 9 T^{2} + p^{2} T^{4}$$
67$D_{4}$ $$1 + T + 46 T^{2} + p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 15 T + 127 T^{2} + 15 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
89$D_{4}$ $$1 + 10 T + 52 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}