Properties

 Degree 4 Conductor 353 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

Origins

Dirichlet series

 L(s)  = 1 − 2-s − 2·3-s − 2·4-s + 5-s + 2·6-s + 3·8-s − 10-s + 2·11-s + 4·12-s − 13-s − 2·15-s + 16-s + 6·17-s − 6·19-s − 2·20-s − 2·22-s − 2·23-s − 6·24-s − 25-s + 26-s + 2·27-s − 2·29-s + 2·30-s − 4·31-s − 2·32-s − 4·33-s − 6·34-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.15·3-s − 4-s + 0.447·5-s + 0.816·6-s + 1.06·8-s − 0.316·10-s + 0.603·11-s + 1.15·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.447·20-s − 0.426·22-s − 0.417·23-s − 1.22·24-s − 1/5·25-s + 0.196·26-s + 0.384·27-s − 0.371·29-s + 0.365·30-s − 0.718·31-s − 0.353·32-s − 0.696·33-s − 1.02·34-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$4$$ $$N$$ = $$353$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{353} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 353,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.1859131786$ $L(\frac12)$ $\approx$ $0.1859131786$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 353$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 353$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad353$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 9 T + p T^{2} )$$
good2$D_{4}$ $$1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4}$$
3$D_{4}$ $$1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
5$D_{4}$ $$1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4}$$
7$V_4$ $$1 - 6 T^{2} + p^{2} T^{4}$$
11$D_{4}$ $$1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} )$$
19$D_{4}$ $$1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
31$D_{4}$ $$1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
41$D_{4}$ $$1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
47$V_4$ $$1 + 23 T^{2} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$V_4$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
89$D_{4}$ $$1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 11 T + 116 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}