# Properties

 Label 2-24e2-8.5-c3-0-13 Degree $2$ Conductor $576$ Sign $0.707 - 0.707i$ Analytic cond. $33.9851$ Root an. cond. $5.82967$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 31.1·7-s + 62.3i·13-s + 56i·19-s + 125·25-s − 155.·31-s − 436. i·37-s + 520i·43-s + 629·49-s + 935. i·61-s + 880i·67-s + 1.19e3·73-s + 1.09e3·79-s + 1.94e3i·91-s + 1.33e3·97-s − 1.02e3·103-s + ⋯
 L(s)  = 1 + 1.68·7-s + 1.33i·13-s + 0.676i·19-s + 25-s − 0.903·31-s − 1.93i·37-s + 1.84i·43-s + 1.83·49-s + 1.96i·61-s + 1.60i·67-s + 1.90·73-s + 1.55·79-s + 2.23i·91-s + 1.39·97-s − 0.984·103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $0.707 - 0.707i$ Analytic conductor: $$33.9851$$ Root analytic conductor: $$5.82967$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{576} (289, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 576,\ (\ :3/2),\ 0.707 - 0.707i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.376640124$$ $$L(\frac12)$$ $$\approx$$ $$2.376640124$$ $$L(\frac{5}{2})$$ 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 - 125T^{2}$$
7 $$1 - 31.1T + 343T^{2}$$
11 $$1 - 1.33e3T^{2}$$
13 $$1 - 62.3iT - 2.19e3T^{2}$$
17 $$1 + 4.91e3T^{2}$$
19 $$1 - 56iT - 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 + 155.T + 2.97e4T^{2}$$
37 $$1 + 436. iT - 5.06e4T^{2}$$
41 $$1 + 6.89e4T^{2}$$
43 $$1 - 520iT - 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 1.48e5T^{2}$$
59 $$1 - 2.05e5T^{2}$$
61 $$1 - 935. iT - 2.26e5T^{2}$$
67 $$1 - 880iT - 3.00e5T^{2}$$
71 $$1 + 3.57e5T^{2}$$
73 $$1 - 1.19e3T + 3.89e5T^{2}$$
79 $$1 - 1.09e3T + 4.93e5T^{2}$$
83 $$1 - 5.71e5T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 - 1.33e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$