# Properties

 Degree $2$ Conductor $1764$ Sign $0.605 - 0.795i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 0.999i·8-s + (−1.73 + i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (1.73 + i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.499i)32-s + (−1 + 1.73i)37-s + (−1.73 − 0.999i)44-s + (−0.999 − 1.73i)46-s − 0.999i·50-s − 0.999·64-s + 2i·71-s + ⋯
 L(s)  = 1 + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 0.999i·8-s + (−1.73 + i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (1.73 + i)23-s + (0.5 + 0.866i)25-s + (0.866 − 0.499i)32-s + (−1 + 1.73i)37-s + (−1.73 − 0.999i)44-s + (−0.999 − 1.73i)46-s − 0.999i·50-s − 0.999·64-s + 2i·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $0.605 - 0.795i$ Motivic weight: $$0$$ Character: $\chi_{1764} (667, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :0),\ 0.605 - 0.795i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5869705535$$ $$L(\frac12)$$ $$\approx$$ $$0.5869705535$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

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