# Properties

 Degree 4 Conductor $7 \cdot 61$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s + 4-s − 5-s − 4·9-s + 2·10-s + 11-s + 3·13-s + 16-s + 8·18-s + 2·19-s − 20-s − 2·22-s + 23-s + 5·25-s − 6·26-s − 4·29-s − 6·31-s + 2·32-s − 4·36-s − 10·37-s − 4·38-s − 3·41-s + 8·43-s + 44-s + 4·45-s − 2·46-s − 6·49-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/2·4-s − 0.447·5-s − 4/3·9-s + 0.632·10-s + 0.301·11-s + 0.832·13-s + 1/4·16-s + 1.88·18-s + 0.458·19-s − 0.223·20-s − 0.426·22-s + 0.208·23-s + 25-s − 1.17·26-s − 0.742·29-s − 1.07·31-s + 0.353·32-s − 2/3·36-s − 1.64·37-s − 0.648·38-s − 0.468·41-s + 1.21·43-s + 0.150·44-s + 0.596·45-s − 0.294·46-s − 6/7·49-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$427$$    =    $$7 \cdot 61$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{427} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 427,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.1899303686$ $L(\frac12)$ $\approx$ $0.1899303686$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{7,\;61\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;61\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_1$$\times$$C_2$ $$( 1 + T )( 1 - T + p T^{2} )$$
61$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 8 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$V_4$ $$1 + 4 T^{2} + p^{2} T^{4}$$
5$V_4$ $$1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
13$D_{4}$ $$1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
17$V_4$ $$1 + 4 T^{2} + p^{2} T^{4}$$
19$D_{4}$ $$1 - 2 T - 2 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
31$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
37$D_{4}$ $$1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$D_{4}$ $$1 + 5 T - 2 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 - 7 T + 18 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + T - 50 T^{2} + p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}