# Properties

 Degree 4 Conductor $2^{6} \cdot 3^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s − 4·5-s + 3·9-s + 8·11-s − 4·13-s + 8·15-s + 4·17-s − 8·19-s − 16·23-s + 2·25-s − 4·27-s + 12·29-s + 16·31-s − 16·33-s + 12·37-s + 8·39-s − 12·41-s + 8·43-s − 12·45-s − 14·49-s − 8·51-s − 4·53-s − 32·55-s + 16·57-s + 8·59-s − 4·61-s + 16·65-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1.78·5-s + 9-s + 2.41·11-s − 1.10·13-s + 2.06·15-s + 0.970·17-s − 1.83·19-s − 3.33·23-s + 2/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s − 2.78·33-s + 1.97·37-s + 1.28·39-s − 1.87·41-s + 1.21·43-s − 1.78·45-s − 2·49-s − 1.12·51-s − 0.549·53-s − 4.31·55-s + 2.11·57-s + 1.04·59-s − 0.512·61-s + 1.98·65-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$576$$    =    $$2^{6} \cdot 3^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{576} (1, \cdot )$ Sato-Tate : $E_1$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 576,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.2906599836$ $L(\frac12)$ $\approx$ $0.2906599836$ $L(\frac{3}{2})$ 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$C_1$ $$( 1 + T )^{2}$$
good5$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
23$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
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}