# Properties

 Degree $2$ Conductor $1764$ Sign $-0.577 - 0.816i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41i·2-s − 2.00·4-s − 2i·5-s − 2.82i·8-s + 2.82·10-s − 7.07·13-s + 4.00·16-s + 2i·17-s + 4.00i·20-s + 25-s − 10.0i·26-s + 9.89i·29-s + 5.65i·32-s − 2.82·34-s + 12·37-s + ⋯
 L(s)  = 1 + 0.999i·2-s − 1.00·4-s − 0.894i·5-s − 1.00i·8-s + 0.894·10-s − 1.96·13-s + 1.00·16-s + 0.485i·17-s + 0.894i·20-s + 0.200·25-s − 1.96i·26-s + 1.83i·29-s + 1.00i·32-s − 0.485·34-s + 1.97·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-0.577 - 0.816i$ Motivic weight: $$1$$ Character: $\chi_{1764} (1079, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :1/2),\ -0.577 - 0.816i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9849348543$$ $$L(\frac12)$$ $$\approx$$ $$0.9849348543$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 1.41iT$$
3 $$1$$
7 $$1$$
good5 $$1 + 2iT - 5T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 + 7.07T + 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 9.89iT - 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 - 12T + 37T^{2}$$
41 $$1 - 8iT - 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 12.7iT - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 15.5T + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 7.07T + 73T^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 - 16iT - 89T^{2}$$
97 $$1 + 18.3T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$