# Properties

 Degree $2$ Conductor $1764$ Sign $-0.676 - 0.736i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.41 + 0.0900i)2-s + (1.98 − 0.254i)4-s + 2.48i·5-s + (−2.77 + 0.537i)8-s + (−0.224 − 3.51i)10-s + 4.60·11-s − 5.22·13-s + (3.87 − 1.00i)16-s + 5.61i·17-s − 3.19i·19-s + (0.632 + 4.93i)20-s + (−6.50 + 0.414i)22-s + 0.718·23-s − 1.19·25-s + (7.37 − 0.470i)26-s + ⋯
 L(s)  = 1 + (−0.997 + 0.0636i)2-s + (0.991 − 0.127i)4-s + 1.11i·5-s + (−0.981 + 0.189i)8-s + (−0.0708 − 1.11i)10-s + 1.38·11-s − 1.44·13-s + (0.967 − 0.252i)16-s + 1.36i·17-s − 0.732i·19-s + (0.141 + 1.10i)20-s + (−1.38 + 0.0884i)22-s + 0.149·23-s − 0.238·25-s + (1.44 − 0.0921i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $-0.676 - 0.736i$ Motivic weight: $$1$$ Character: $\chi_{1764} (1079, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :1/2),\ -0.676 - 0.736i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8016294474$$ $$L(\frac12)$$ $$\approx$$ $$0.8016294474$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.41 - 0.0900i)T$$
3 $$1$$
7 $$1$$
good5 $$1 - 2.48iT - 5T^{2}$$
11 $$1 - 4.60T + 11T^{2}$$
13 $$1 + 5.22T + 13T^{2}$$
17 $$1 - 5.61iT - 17T^{2}$$
19 $$1 + 3.19iT - 19T^{2}$$
23 $$1 - 0.718T + 23T^{2}$$
29 $$1 - 4.53iT - 29T^{2}$$
31 $$1 + 1.17iT - 31T^{2}$$
37 $$1 - 2.71T + 37T^{2}$$
41 $$1 - 3.83iT - 41T^{2}$$
43 $$1 - 11.1iT - 43T^{2}$$
47 $$1 + 5.41T + 47T^{2}$$
53 $$1 - 2.06iT - 53T^{2}$$
59 $$1 - 4.11T + 59T^{2}$$
61 $$1 - 1.01T + 61T^{2}$$
67 $$1 + 12.6iT - 67T^{2}$$
71 $$1 + 7.31T + 71T^{2}$$
73 $$1 + 9.63T + 73T^{2}$$
79 $$1 - 8.83iT - 79T^{2}$$
83 $$1 + 13.7T + 83T^{2}$$
89 $$1 + 8.52iT - 89T^{2}$$
97 $$1 - 10.7T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$