# Properties

 Degree $2$ Conductor $576$ Sign $-0.479 - 0.877i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−6.49 − 6.49i)5-s − 3.94·7-s + (4.31 − 4.31i)11-s + (4.06 − 4.06i)13-s + 14.5·17-s + (−4.94 − 4.94i)19-s − 43.6·23-s + 59.3i·25-s + (−25.0 + 25.0i)29-s + 32.5i·31-s + (25.6 + 25.6i)35-s + (4.14 + 4.14i)37-s + 55.3i·41-s + (16.1 − 16.1i)43-s + 7.92i·47-s + ⋯
 L(s)  = 1 + (−1.29 − 1.29i)5-s − 0.563·7-s + (0.391 − 0.391i)11-s + (0.312 − 0.312i)13-s + 0.856·17-s + (−0.260 − 0.260i)19-s − 1.89·23-s + 2.37i·25-s + (−0.865 + 0.865i)29-s + 1.04i·31-s + (0.731 + 0.731i)35-s + (0.111 + 0.111i)37-s + 1.34i·41-s + (0.374 − 0.374i)43-s + 0.168i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$576$$    =    $$2^{6} \cdot 3^{2}$$ Sign: $-0.479 - 0.877i$ Motivic weight: $$2$$ Character: $\chi_{576} (271, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 576,\ (\ :1),\ -0.479 - 0.877i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.1334756923$$ $$L(\frac12)$$ $$\approx$$ $$0.1334756923$$ $$L(2)$$ 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 + (6.49 + 6.49i)T + 25iT^{2}$$
7 $$1 + 3.94T + 49T^{2}$$
11 $$1 + (-4.31 + 4.31i)T - 121iT^{2}$$
13 $$1 + (-4.06 + 4.06i)T - 169iT^{2}$$
17 $$1 - 14.5T + 289T^{2}$$
19 $$1 + (4.94 + 4.94i)T + 361iT^{2}$$
23 $$1 + 43.6T + 529T^{2}$$
29 $$1 + (25.0 - 25.0i)T - 841iT^{2}$$
31 $$1 - 32.5iT - 961T^{2}$$
37 $$1 + (-4.14 - 4.14i)T + 1.36e3iT^{2}$$
41 $$1 - 55.3iT - 1.68e3T^{2}$$
43 $$1 + (-16.1 + 16.1i)T - 1.84e3iT^{2}$$
47 $$1 - 7.92iT - 2.20e3T^{2}$$
53 $$1 + (-31.5 - 31.5i)T + 2.80e3iT^{2}$$
59 $$1 + (49.7 - 49.7i)T - 3.48e3iT^{2}$$
61 $$1 + (-44.4 + 44.4i)T - 3.72e3iT^{2}$$
67 $$1 + (-1.64 - 1.64i)T + 4.48e3iT^{2}$$
71 $$1 - 24.1T + 5.04e3T^{2}$$
73 $$1 + 10.7iT - 5.32e3T^{2}$$
79 $$1 + 72.0iT - 6.24e3T^{2}$$
83 $$1 + (-42.0 - 42.0i)T + 6.88e3iT^{2}$$
89 $$1 + 28.9iT - 7.92e3T^{2}$$
97 $$1 + 54.2T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$