# Properties

 Label 2-864-4.3-c2-0-3 Degree $2$ Conductor $864$ Sign $-0.707 - 0.707i$ Analytic cond. $23.5422$ Root an. cond. $4.85204$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.22·5-s − 6.57i·7-s + 13.8i·11-s − 19.0·13-s − 23.0·17-s + 18.8i·19-s − 13.5i·23-s − 14.5·25-s − 0.752·29-s + 46.4i·31-s − 21.1i·35-s − 2.68·37-s + 34.1·41-s + 20.9i·43-s + 15.4i·47-s + ⋯
 L(s)  = 1 + 0.645·5-s − 0.938i·7-s + 1.26i·11-s − 1.46·13-s − 1.35·17-s + 0.991i·19-s − 0.591i·23-s − 0.583·25-s − 0.0259·29-s + 1.49i·31-s − 0.605i·35-s − 0.0724·37-s + 0.832·41-s + 0.486i·43-s + 0.327i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$864$$    =    $$2^{5} \cdot 3^{3}$$ Sign: $-0.707 - 0.707i$ Analytic conductor: $$23.5422$$ Root analytic conductor: $$4.85204$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{864} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 864,\ (\ :1),\ -0.707 - 0.707i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.6710884168$$ $$L(\frac12)$$ $$\approx$$ $$0.6710884168$$ $$L(2)$$ 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 - 3.22T + 25T^{2}$$
7 $$1 + 6.57iT - 49T^{2}$$
11 $$1 - 13.8iT - 121T^{2}$$
13 $$1 + 19.0T + 169T^{2}$$
17 $$1 + 23.0T + 289T^{2}$$
19 $$1 - 18.8iT - 361T^{2}$$
23 $$1 + 13.5iT - 529T^{2}$$
29 $$1 + 0.752T + 841T^{2}$$
31 $$1 - 46.4iT - 961T^{2}$$
37 $$1 + 2.68T + 1.36e3T^{2}$$
41 $$1 - 34.1T + 1.68e3T^{2}$$
43 $$1 - 20.9iT - 1.84e3T^{2}$$
47 $$1 - 15.4iT - 2.20e3T^{2}$$
53 $$1 - 46.8T + 2.80e3T^{2}$$
59 $$1 - 40.4iT - 3.48e3T^{2}$$
61 $$1 + 105.T + 3.72e3T^{2}$$
67 $$1 - 27.9iT - 4.48e3T^{2}$$
71 $$1 + 24.0iT - 5.04e3T^{2}$$
73 $$1 + 120.T + 5.32e3T^{2}$$
79 $$1 - 95.6iT - 6.24e3T^{2}$$
83 $$1 + 115. iT - 6.88e3T^{2}$$
89 $$1 + 169.T + 7.92e3T^{2}$$
97 $$1 + 93.1T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$