# Properties

 Degree 4 Conductor $2^{4} \cdot 3 \cdot 23$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 4·5-s − 2·9-s + 2·11-s + 13-s − 2·17-s + 2·19-s + 3·23-s + 6·25-s + 3·27-s − 29-s + 3·31-s − 12·37-s + 41-s − 4·43-s + 8·45-s + 3·47-s + 10·49-s − 4·53-s − 8·55-s − 8·59-s − 8·61-s − 4·65-s + 2·67-s + 7·71-s + 11·73-s − 10·79-s + 4·81-s + ⋯
 L(s)  = 1 − 1.78·5-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s + 0.625·23-s + 6/5·25-s + 0.577·27-s − 0.185·29-s + 0.538·31-s − 1.97·37-s + 0.156·41-s − 0.609·43-s + 1.19·45-s + 0.437·47-s + 10/7·49-s − 0.549·53-s − 1.07·55-s − 1.04·59-s − 1.02·61-s − 0.496·65-s + 0.244·67-s + 0.830·71-s + 1.28·73-s − 1.12·79-s + 4/9·81-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1104$$    =    $$2^{4} \cdot 3 \cdot 23$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1104} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 1104,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.4453748542$ $L(\frac12)$ $\approx$ $0.4453748542$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;23\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;23\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
3$C_1$$\times$$C_2$ $$( 1 - T )( 1 + T + p T^{2} )$$
23$C_1$$\times$$C_2$ $$( 1 + T )( 1 - 4 T + p T^{2} )$$
good5$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$V_4$ $$1 - 10 T^{2} + p^{2} T^{4}$$
11$V_4$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$D_{4}$ $$1 - T - p T^{3} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} )$$
53$D_{4}$ $$1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
71$C_2$$\times$$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$D_{4}$ $$1 - 11 T + 124 T^{2} - 11 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
89$D_{4}$ $$1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}