# Properties

 Degree 4 Conductor $3 \cdot 5 \cdot 23 \cdot 29$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 2

# Origins

## Dirichlet series

 L(s)  = 1 − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 5·7-s − 3·8-s + 4·9-s + 9·10-s − 7·11-s − 8·12-s − 4·13-s + 15·14-s + 6·15-s + 3·16-s − 17-s − 12·18-s − 2·19-s − 12·20-s + 10·21-s + 21·22-s − 7·23-s + 6·24-s + 2·25-s + 12·26-s − 5·27-s − 20·28-s + ⋯
 L(s)  = 1 − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 1.88·7-s − 1.06·8-s + 4/3·9-s + 2.84·10-s − 2.11·11-s − 2.30·12-s − 1.10·13-s + 4.00·14-s + 1.54·15-s + 3/4·16-s − 0.242·17-s − 2.82·18-s − 0.458·19-s − 2.68·20-s + 2.18·21-s + 4.47·22-s − 1.45·23-s + 1.22·24-s + 2/5·25-s + 2.35·26-s − 0.962·27-s − 3.77·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$10005$$    =    $$3 \cdot 5 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{10005} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 2 Selberg data = $(4,\ 10005,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5,\;23,\;29\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;23,\;29\}$, 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$ $$( 1 - T )( 1 + p T + p T^{2} )$$
5$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 2 T + p T^{2} )$$
23$C_1$$\times$$C_2$ $$( 1 + T )( 1 + 6 T + p T^{2} )$$
29$C_1$$\times$$C_2$ $$( 1 - T )( 1 + 3 T + p T^{2} )$$
good2$C_2^2$ $$1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
7$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 5 T + p T^{2} )$$
11$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
13$D_{4}$ $$1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4}$$
19$C_2^2$ $$1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - T - 8 T^{2} - p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4}$$
43$D_{4}$ $$1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
47$D_{4}$ $$1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2^2$ $$1 - 90 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
67$D_{4}$ $$1 + 5 T + 112 T^{2} + 5 p T^{3} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + p T^{2} )$$
73$D_{4}$ $$1 - 5 T + 56 T^{2} - 5 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 18 T + 166 T^{2} + 18 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 + 17 T + 188 T^{2} + 17 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 6 T - 42 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}