# Properties

 Label 2-24e2-1.1-c3-0-20 Degree $2$ Conductor $576$ Sign $-1$ Analytic cond. $33.9851$ Root an. cond. $5.82967$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 20·7-s + 70·13-s + 56·19-s − 125·25-s − 308·31-s − 110·37-s − 520·43-s + 57·49-s − 182·61-s − 880·67-s + 1.19e3·73-s − 884·79-s − 1.40e3·91-s − 1.33e3·97-s − 1.82e3·103-s + 646·109-s + ⋯
 L(s)  = 1 − 1.07·7-s + 1.49·13-s + 0.676·19-s − 25-s − 1.78·31-s − 0.488·37-s − 1.84·43-s + 0.166·49-s − 0.382·61-s − 1.60·67-s + 1.90·73-s − 1.25·79-s − 1.61·91-s − 1.39·97-s − 1.74·103-s + 0.567·109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $-1$ Analytic conductor: $$33.9851$$ Root analytic conductor: $$5.82967$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 576,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + p^{3} T^{2}$$
7 $$1 + 20 T + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 - 70 T + p^{3} T^{2}$$
17 $$1 + p^{3} T^{2}$$
19 $$1 - 56 T + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + p^{3} T^{2}$$
31 $$1 + 308 T + p^{3} T^{2}$$
37 $$1 + 110 T + p^{3} T^{2}$$
41 $$1 + p^{3} T^{2}$$
43 $$1 + 520 T + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 + 182 T + p^{3} T^{2}$$
67 $$1 + 880 T + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 - 1190 T + p^{3} T^{2}$$
79 $$1 + 884 T + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 + p^{3} T^{2}$$
97 $$1 + 1330 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$