# Properties

 Label 2-24e2-8.5-c3-0-21 Degree $2$ Conductor $576$ Sign $0.258 + 0.965i$ 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 + 13.8i·5-s + 3.46·7-s − 48i·11-s − 20.7i·13-s − 96·17-s − 40i·19-s − 110.·23-s − 66.9·25-s + 13.8i·29-s + 204.·31-s + 47.9i·35-s − 297. i·37-s + 288·41-s − 152i·43-s + 554.·47-s + ⋯
 L(s)  = 1 + 1.23i·5-s + 0.187·7-s − 1.31i·11-s − 0.443i·13-s − 1.36·17-s − 0.482i·19-s − 1.00·23-s − 0.535·25-s + 0.0887i·29-s + 1.18·31-s + 0.231i·35-s − 1.32i·37-s + 1.09·41-s − 0.539i·43-s + 1.72·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $0.258 + 0.965i$ 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: $$(2,\ 576,\ (\ :3/2),\ 0.258 + 0.965i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.263682639$$ $$L(\frac12)$$ $$\approx$$ $$1.263682639$$ $$L(\frac{5}{2})$$ 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 - 13.8iT - 125T^{2}$$
7 $$1 - 3.46T + 343T^{2}$$
11 $$1 + 48iT - 1.33e3T^{2}$$
13 $$1 + 20.7iT - 2.19e3T^{2}$$
17 $$1 + 96T + 4.91e3T^{2}$$
19 $$1 + 40iT - 6.85e3T^{2}$$
23 $$1 + 110.T + 1.21e4T^{2}$$
29 $$1 - 13.8iT - 2.43e4T^{2}$$
31 $$1 - 204.T + 2.97e4T^{2}$$
37 $$1 + 297. iT - 5.06e4T^{2}$$
41 $$1 - 288T + 6.89e4T^{2}$$
43 $$1 + 152iT - 7.95e4T^{2}$$
47 $$1 - 554.T + 1.03e5T^{2}$$
53 $$1 + 180. iT - 1.48e5T^{2}$$
59 $$1 - 480iT - 2.05e5T^{2}$$
61 $$1 + 755. iT - 2.26e5T^{2}$$
67 $$1 + 848iT - 3.00e5T^{2}$$
71 $$1 + 886.T + 3.57e5T^{2}$$
73 $$1 + 538T + 3.89e5T^{2}$$
79 $$1 - 1.00e3T + 4.93e5T^{2}$$
83 $$1 + 432iT - 5.71e5T^{2}$$
89 $$1 + 1.34e3T + 7.04e5T^{2}$$
97 $$1 + 590T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$