# Properties

 Degree 4 Conductor $5 \cdot 101 \cdot 113$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 3

# Origins

## Dirichlet series

 L(s)  = 1 − 3·2-s − 4·3-s + 4·4-s − 3·5-s + 12·6-s − 5·7-s − 3·8-s + 7·9-s + 9·10-s − 5·11-s − 16·12-s − 6·13-s + 15·14-s + 12·15-s + 3·16-s − 5·17-s − 21·18-s − 5·19-s − 12·20-s + 20·21-s + 15·22-s − 9·23-s + 12·24-s + 2·25-s + 18·26-s − 4·27-s − 20·28-s + ⋯
 L(s)  = 1 − 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s + 4.89·6-s − 1.88·7-s − 1.06·8-s + 7/3·9-s + 2.84·10-s − 1.50·11-s − 4.61·12-s − 1.66·13-s + 4.00·14-s + 3.09·15-s + 3/4·16-s − 1.21·17-s − 4.94·18-s − 1.14·19-s − 2.68·20-s + 4.36·21-s + 3.19·22-s − 1.87·23-s + 2.44·24-s + 2/5·25-s + 3.53·26-s − 0.769·27-s − 3.77·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 57065 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 57065 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$57065$$    =    $$5 \cdot 101 \cdot 113$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{57065} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 3 Selberg data = $(4,\ 57065,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;101,\;113\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;101,\;113\}$, 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$ $$( 1 + T )( 1 + 2 T + p T^{2} )$$
101$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 8 T + p T^{2} )$$
113$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 2 T + p T^{2} )$$
good2$C_2^2$ $$1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
3$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + p T + p T^{2} )$$
7$D_{4}$ $$1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + T - 37 T^{2} + p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 9 T + 77 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - T - 14 T^{2} - p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 9 T + 80 T^{2} + 9 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 8 T + 13 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$D_{4}$ $$1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 10 T + 100 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}