# Properties

 Degree 4 Conductor $2^{7} \cdot 3 \cdot 19$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s + 8·11-s − 12-s + 16-s − 8·17-s − 2·18-s − 19-s + 8·22-s − 24-s − 2·25-s + 2·27-s + 32-s − 8·33-s − 8·34-s − 2·36-s − 38-s − 4·41-s − 4·43-s + 8·44-s − 48-s + 2·49-s − 2·50-s + ⋯
 L(s)  = 1 + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 2.41·11-s − 0.288·12-s + 1/4·16-s − 1.94·17-s − 0.471·18-s − 0.229·19-s + 1.70·22-s − 0.204·24-s − 2/5·25-s + 0.384·27-s + 0.176·32-s − 1.39·33-s − 1.37·34-s − 1/3·36-s − 0.162·38-s − 0.624·41-s − 0.609·43-s + 1.20·44-s − 0.144·48-s + 2/7·49-s − 0.282·50-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 7296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 7296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$7296$$    =    $$2^{7} \cdot 3 \cdot 19$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{7296} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 7296,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $1.156426687$ $L(\frac12)$ $\approx$ $1.156426687$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;19\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ $$1 - T$$
3$C_1$$\times$$C_2$ $$( 1 + T )( 1 + p T^{2} )$$
19$C_1$$\times$$C_2$ $$( 1 + T )( 1 + p T^{2} )$$
good5$V_4$ $$1 + 2 T^{2} + p^{2} T^{4}$$
7$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
11$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
13$V_4$ $$1 - 6 T^{2} + p^{2} T^{4}$$
17$C_2$$\times$$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
29$V_4$ $$1 - 10 T^{2} + p^{2} T^{4}$$
31$V_4$ $$1 - 34 T^{2} + p^{2} T^{4}$$
37$V_4$ $$1 + 42 T^{2} + p^{2} T^{4}$$
41$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$V_4$ $$1 - 58 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$V_4$ $$1 + 62 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 + 14 T + p T^{2} )^{2}$$
79$V_4$ $$1 + 30 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}