# Properties

 Label 2-24e2-3.2-c6-0-2 Degree $2$ Conductor $576$ Sign $0.577 + 0.816i$ Analytic cond. $132.511$ Root an. cond. $11.5113$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 173. i·5-s − 484·7-s + 1.34e3i·11-s − 3.36e3·13-s − 12.7i·17-s − 5.74e3·19-s + 3.37e3i·23-s − 1.46e4·25-s + 2.93e4i·29-s − 3.97e4·31-s − 8.41e4i·35-s − 5.25e4·37-s + 3.70e4i·41-s − 3.80e3·43-s − 7.67e4i·47-s + ⋯
 L(s)  = 1 + 1.39i·5-s − 1.41·7-s + 1.00i·11-s − 1.53·13-s − 0.00259i·17-s − 0.837·19-s + 0.277i·23-s − 0.936·25-s + 1.20i·29-s − 1.33·31-s − 1.96i·35-s − 1.03·37-s + 0.537i·41-s − 0.0477·43-s − 0.739i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $0.577 + 0.816i$ Analytic conductor: $$132.511$$ Root analytic conductor: $$11.5113$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{576} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 576,\ (\ :3),\ 0.577 + 0.816i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.1320302035$$ $$L(\frac12)$$ $$\approx$$ $$0.1320302035$$ $$L(4)$$ 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 - 173. iT - 1.56e4T^{2}$$
7 $$1 + 484T + 1.17e5T^{2}$$
11 $$1 - 1.34e3iT - 1.77e6T^{2}$$
13 $$1 + 3.36e3T + 4.82e6T^{2}$$
17 $$1 + 12.7iT - 2.41e7T^{2}$$
19 $$1 + 5.74e3T + 4.70e7T^{2}$$
23 $$1 - 3.37e3iT - 1.48e8T^{2}$$
29 $$1 - 2.93e4iT - 5.94e8T^{2}$$
31 $$1 + 3.97e4T + 8.87e8T^{2}$$
37 $$1 + 5.25e4T + 2.56e9T^{2}$$
41 $$1 - 3.70e4iT - 4.75e9T^{2}$$
43 $$1 + 3.80e3T + 6.32e9T^{2}$$
47 $$1 + 7.67e4iT - 1.07e10T^{2}$$
53 $$1 - 2.38e5iT - 2.21e10T^{2}$$
59 $$1 + 2.49e5iT - 4.21e10T^{2}$$
61 $$1 + 1.32e4T + 5.15e10T^{2}$$
67 $$1 + 1.68e5T + 9.04e10T^{2}$$
71 $$1 - 5.31e5iT - 1.28e11T^{2}$$
73 $$1 - 2.36e5T + 1.51e11T^{2}$$
79 $$1 + 3.51e4T + 2.43e11T^{2}$$
83 $$1 + 1.09e4iT - 3.26e11T^{2}$$
89 $$1 - 1.29e5iT - 4.96e11T^{2}$$
97 $$1 + 3.21e5T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$