# Properties

 Label 2-648-216.43-c0-0-0 Degree $2$ Conductor $648$ Sign $0.835 + 0.549i$ Analytic cond. $0.323394$ Root an. cond. $0.568677$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.500 − 0.866i)8-s + (0.326 − 0.118i)11-s + (−0.939 + 0.342i)16-s + (0.766 + 1.32i)17-s + (0.939 − 1.62i)19-s + (−0.326 − 0.118i)22-s + (0.766 + 0.642i)25-s + (0.939 + 0.342i)32-s + (0.266 − 1.50i)34-s + (−1.76 + 0.642i)38-s + (1.43 − 1.20i)41-s + (−1.43 + 0.524i)43-s + (0.173 + 0.300i)44-s + ⋯
 L(s)  = 1 + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.500 − 0.866i)8-s + (0.326 − 0.118i)11-s + (−0.939 + 0.342i)16-s + (0.766 + 1.32i)17-s + (0.939 − 1.62i)19-s + (−0.326 − 0.118i)22-s + (0.766 + 0.642i)25-s + (0.939 + 0.342i)32-s + (0.266 − 1.50i)34-s + (−1.76 + 0.642i)38-s + (1.43 − 1.20i)41-s + (−1.43 + 0.524i)43-s + (0.173 + 0.300i)44-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$648$$    =    $$2^{3} \cdot 3^{4}$$ Sign: $0.835 + 0.549i$ Analytic conductor: $$0.323394$$ Root analytic conductor: $$0.568677$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{648} (451, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 648,\ (\ :0),\ 0.835 + 0.549i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6843311926$$ $$L(\frac12)$$ $$\approx$$ $$0.6843311926$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.766 + 0.642i)T$$
3 $$1$$
good5 $$1 + (-0.766 - 0.642i)T^{2}$$
7 $$1 + (0.939 + 0.342i)T^{2}$$
11 $$1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2}$$
13 $$1 + (-0.173 + 0.984i)T^{2}$$
17 $$1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2}$$
19 $$1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (0.939 - 0.342i)T^{2}$$
29 $$1 + (-0.173 - 0.984i)T^{2}$$
31 $$1 + (0.939 - 0.342i)T^{2}$$
37 $$1 + (0.5 - 0.866i)T^{2}$$
41 $$1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2}$$
43 $$1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2}$$
47 $$1 + (0.939 + 0.342i)T^{2}$$
53 $$1 - T^{2}$$
59 $$1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2}$$
61 $$1 + (0.939 + 0.342i)T^{2}$$
67 $$1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2}$$
71 $$1 + (0.5 - 0.866i)T^{2}$$
73 $$1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2}$$
79 $$1 + (-0.173 - 0.984i)T^{2}$$
83 $$1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}$$
89 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
97 $$1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$