# Properties

 Degree 4 Conductor $3 \cdot 359$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·3-s + 2·5-s − 3·7-s − 11-s + 13-s − 4·15-s − 4·16-s + 2·17-s − 19-s + 6·21-s + 4·23-s − 2·25-s + 5·27-s + 6·29-s − 4·31-s + 2·33-s − 6·35-s + 9·37-s − 2·39-s + 6·41-s + 2·43-s + 2·47-s + 8·48-s + 3·49-s − 4·51-s − 5·53-s − 2·55-s + ⋯
 L(s)  = 1 − 1.15·3-s + 0.894·5-s − 1.13·7-s − 0.301·11-s + 0.277·13-s − 1.03·15-s − 16-s + 0.485·17-s − 0.229·19-s + 1.30·21-s + 0.834·23-s − 2/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s − 1.01·35-s + 1.47·37-s − 0.320·39-s + 0.937·41-s + 0.304·43-s + 0.291·47-s + 1.15·48-s + 3/7·49-s − 0.560·51-s − 0.686·53-s − 0.269·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1077$$    =    $$3 \cdot 359$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1077} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 1077,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.4062914313$ $L(\frac12)$ $\approx$ $0.4062914313$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;359\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;359\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_2$ $$( 1 + T )( 1 + T + p T^{2} )$$
359$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 30 T + p T^{2} )$$
good2$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
5$D_{4}$ $$1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
13$C_2$$\times$$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$D_{4}$ $$1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
23$D_{4}$ $$1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + p T^{2} )$$
31$D_{4}$ $$1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 2 T - 52 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
47$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
53$D_{4}$ $$1 + 5 T + 75 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 14 T + 160 T^{2} + 14 p T^{3} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_4$ $$1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 3 T - 27 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 3 T + 78 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 5 T + 3 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}