# Properties

 Degree $2$ Conductor $1764$ Sign $0.755 - 0.654i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.73i·13-s + 1.73i·19-s + 25-s − 1.73i·31-s + 37-s + 43-s − 67-s + 1.73i·73-s − 79-s − 1.73i·103-s + 109-s + ⋯
 L(s)  = 1 + 1.73i·13-s + 1.73i·19-s + 25-s − 1.73i·31-s + 37-s + 43-s − 67-s + 1.73i·73-s − 79-s − 1.73i·103-s + 109-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $0.755 - 0.654i$ Motivic weight: $$0$$ Character: $\chi_{1764} (685, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :0),\ 0.755 - 0.654i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.118390064$$ $$L(\frac12)$$ $$\approx$$ $$1.118390064$$ $$L(1)$$ 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$$
good5 $$1 - T^{2}$$
11 $$1 + T^{2}$$
13 $$1 - 1.73iT - T^{2}$$
17 $$1 - T^{2}$$
19 $$1 - 1.73iT - T^{2}$$
23 $$1 + T^{2}$$
29 $$1 + T^{2}$$
31 $$1 + 1.73iT - T^{2}$$
37 $$1 - T + T^{2}$$
41 $$1 - T^{2}$$
43 $$1 - T + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + T + T^{2}$$
71 $$1 + T^{2}$$
73 $$1 - 1.73iT - T^{2}$$
79 $$1 + T + T^{2}$$
83 $$1 - T^{2}$$
89 $$1 - T^{2}$$
97 $$1 - T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−9.546241743492637231301743564817, −8.865857949801841562228746163395, −7.990575680104474115606634308158, −7.22425857804538120847758920325, −6.34061604149370236202950670022, −5.67759528043369136618398569354, −4.43807658020438802710731254167, −3.91280339917285512320307417121, −2.57510274776060613681995637119, −1.50487015914138607533452316670, 0.942561592259345294468296927486, 2.63217218719953035275204927225, 3.26571334090267834472061865824, 4.61772661579888641664611156362, 5.24536026167022527772398936435, 6.19813868057511161059301623153, 7.08962852102827203835123985182, 7.79251351648400939042685222173, 8.694103321015126369281764338354, 9.257921809794012541669276835668