# Properties

 Degree 4 Conductor $2^{8} \cdot 3^{3}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 3-s + 9-s − 8·19-s + 2·25-s + 27-s − 8·43-s + 2·49-s − 8·57-s − 8·67-s + 4·73-s + 2·75-s + 81-s + 4·97-s − 22·121-s + 127-s − 8·129-s + 131-s + 137-s + 139-s + 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 8·171-s + ⋯
 L(s)  = 1 + 0.577·3-s + 1/3·9-s − 1.83·19-s + 2/5·25-s + 0.192·27-s − 1.21·43-s + 2/7·49-s − 1.05·57-s − 0.977·67-s + 0.468·73-s + 0.230·75-s + 1/9·81-s + 0.406·97-s − 2·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.164·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 0.611·171-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$6912$$    =    $$2^{8} \cdot 3^{3}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{6912} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 6912,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $1.052398998$ $L(\frac12)$ $\approx$ $1.052398998$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 $$1$$
3$C_1$ $$1 - T$$
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$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - p T^{2} )^{2}$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
23$V_4$ $$1 - 2 T^{2} + p^{2} T^{4}$$
29$V_4$ $$1 + 46 T^{2} + p^{2} T^{4}$$
31$V_4$ $$1 - 50 T^{2} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
47$V_4$ $$1 + 46 T^{2} + p^{2} T^{4}$$
53$V_4$ $$1 + 94 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$V_4$ $$1 + 94 T^{2} + p^{2} T^{4}$$
73$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
79$V_4$ $$1 - 50 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}