# Properties

 Degree 4 Conductor $139 \cdot 449$ 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 − 5·5-s + 6·6-s − 5·7-s + 4·8-s + 2·9-s + 10·10-s − 11-s − 9·13-s + 10·14-s + 15·15-s − 4·16-s − 3·17-s − 4·18-s − 11·19-s + 15·21-s + 2·22-s + 6·23-s − 12·24-s + 10·25-s + 18·26-s + 6·27-s − 30·30-s + 3·33-s + 6·34-s + 25·35-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s − 2.23·5-s + 2.44·6-s − 1.88·7-s + 1.41·8-s + 2/3·9-s + 3.16·10-s − 0.301·11-s − 2.49·13-s + 2.67·14-s + 3.87·15-s − 16-s − 0.727·17-s − 0.942·18-s − 2.52·19-s + 3.27·21-s + 0.426·22-s + 1.25·23-s − 2.44·24-s + 2·25-s + 3.53·26-s + 1.15·27-s − 5.47·30-s + 0.522·33-s + 1.02·34-s + 4.22·35-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 62411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 62411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$62411$$    =    $$139 \cdot 449$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{62411} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 3 Selberg data = $(4,\ 62411,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{139,\;449\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{139,\;449\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad139$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 19 T + p T^{2} )$$
449$C_1$$\times$$C_2$ $$( 1 - T )( 1 - 5 T + p T^{2} )$$
good2$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + p T + p T^{2} )$$
3$D_{4}$ $$1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5$C_4$ $$1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 5 T + p T^{2} )$$
11$D_{4}$ $$1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4}$$
13$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
17$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$D_{4}$ $$1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
23$D_{4}$ $$1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 8 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 14 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
41$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
43$D_{4}$ $$1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 5 T + 22 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
71$D_{4}$ $$1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 + 21 T + 240 T^{2} + 21 p T^{3} + p^{2} T^{4}$$
79$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 5 T + 190 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}