# Properties

 Label 2-2268-63.2-c0-0-1 Degree $2$ Conductor $2268$ Sign $0.805 + 0.592i$ Analytic cond. $1.13187$ Root an. cond. $1.06389$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 7-s + (−1 − 1.73i)13-s + (0.5 − 0.866i)19-s + 25-s + (0.5 − 0.866i)31-s + (−1 + 1.73i)37-s + (0.5 − 0.866i)43-s + 49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (−1 − 1.73i)91-s + (0.5 − 0.866i)97-s + 2·103-s + ⋯
 L(s)  = 1 + 7-s + (−1 − 1.73i)13-s + (0.5 − 0.866i)19-s + 25-s + (0.5 − 0.866i)31-s + (−1 + 1.73i)37-s + (0.5 − 0.866i)43-s + 49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (−1 − 1.73i)91-s + (0.5 − 0.866i)97-s + 2·103-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2268$$    =    $$2^{2} \cdot 3^{4} \cdot 7$$ Sign: $0.805 + 0.592i$ Analytic conductor: $$1.13187$$ Root analytic conductor: $$1.06389$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2268} (1241, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2268,\ (\ :0),\ 0.805 + 0.592i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.277778509$$ $$L(\frac12)$$ $$\approx$$ $$1.277778509$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 - T$$
good5 $$1 - T^{2}$$
11 $$1 - T^{2}$$
13 $$1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T^{2}$$
19 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + (0.5 + 0.866i)T^{2}$$
31 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
37 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
41 $$1 + (0.5 - 0.866i)T^{2}$$
43 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
47 $$1 + (0.5 - 0.866i)T^{2}$$
53 $$1 + (0.5 - 0.866i)T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
67 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
79 $$1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}$$
83 $$1 + (0.5 + 0.866i)T^{2}$$
89 $$1 + (0.5 + 0.866i)T^{2}$$
97 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$