# Properties

 Degree 4 Conductor $5^{2} \cdot 4013$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s + 4-s + 3·7-s − 13-s − 6·14-s + 16-s − 8·17-s − 7·19-s + 25-s + 2·26-s + 3·28-s + 6·29-s + 7·31-s + 2·32-s + 16·34-s + 6·37-s + 14·38-s − 3·41-s − 43-s − 47-s + 2·49-s − 2·50-s − 52-s − 2·53-s − 12·58-s − 5·59-s − 7·61-s + ⋯
 L(s)  = 1 − 1.41·2-s + 1/2·4-s + 1.13·7-s − 0.277·13-s − 1.60·14-s + 1/4·16-s − 1.94·17-s − 1.60·19-s + 1/5·25-s + 0.392·26-s + 0.566·28-s + 1.11·29-s + 1.25·31-s + 0.353·32-s + 2.74·34-s + 0.986·37-s + 2.27·38-s − 0.468·41-s − 0.152·43-s − 0.145·47-s + 2/7·49-s − 0.282·50-s − 0.138·52-s − 0.274·53-s − 1.57·58-s − 0.650·59-s − 0.896·61-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$100325$$    =    $$5^{2} \cdot 4013$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{100325} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(4,\ 100325,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;4013\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;4013\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
4013$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 83 T + p T^{2} )$$
good2$D_{4}$ $$1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
3$V_4$ $$1 + p^{2} T^{4}$$
7$D_{4}$ $$1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4}$$
11$V_4$ $$1 + 10 T^{2} + p^{2} T^{4}$$
13$D_{4}$ $$1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
23$V_4$ $$1 + 18 T^{2} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 5 T + 103 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
71$D_{4}$ $$1 + 21 T + 245 T^{2} + 21 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 5 T + 3 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 14 T + 172 T^{2} - 14 p T^{3} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 - 18 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
97$D_{4}$ $$1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}