# Properties

 Degree $2$ Conductor $1764$ Sign $0.654 - 0.755i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + i)2-s − 2i·4-s + 1.73i·5-s + (2 + 2i)8-s + (−1.73 − 1.73i)10-s + i·11-s − 3.46i·13-s − 4·16-s − 1.73i·17-s + 5.19·19-s + 3.46·20-s + (−1 − i)22-s − i·23-s + 2.00·25-s + (3.46 + 3.46i)26-s + ⋯
 L(s)  = 1 + (−0.707 + 0.707i)2-s − i·4-s + 0.774i·5-s + (0.707 + 0.707i)8-s + (−0.547 − 0.547i)10-s + 0.301i·11-s − 0.960i·13-s − 16-s − 0.420i·17-s + 1.19·19-s + 0.774·20-s + (−0.213 − 0.213i)22-s − 0.208i·23-s + 0.400·25-s + (0.679 + 0.679i)26-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$1764$$    =    $$2^{2} \cdot 3^{2} \cdot 7^{2}$$ Sign: $0.654 - 0.755i$ Motivic weight: $$1$$ Character: $\chi_{1764} (1567, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1764,\ (\ :1/2),\ 0.654 - 0.755i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.189606397$$ $$L(\frac12)$$ $$\approx$$ $$1.189606397$$ $$L(\frac{3}{2})$$ 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 - i)T$$
3 $$1$$
7 $$1$$
good5 $$1 - 1.73iT - 5T^{2}$$
11 $$1 - iT - 11T^{2}$$
13 $$1 + 3.46iT - 13T^{2}$$
17 $$1 + 1.73iT - 17T^{2}$$
19 $$1 - 5.19T + 19T^{2}$$
23 $$1 + iT - 23T^{2}$$
29 $$1 + 4T + 29T^{2}$$
31 $$1 - 1.73T + 31T^{2}$$
37 $$1 - 3T + 37T^{2}$$
41 $$1 + 3.46iT - 41T^{2}$$
43 $$1 - 2iT - 43T^{2}$$
47 $$1 - 8.66T + 47T^{2}$$
53 $$1 - T + 53T^{2}$$
59 $$1 + 5.19T + 59T^{2}$$
61 $$1 + 5.19iT - 61T^{2}$$
67 $$1 + 3iT - 67T^{2}$$
71 $$1 - 14iT - 71T^{2}$$
73 $$1 + 8.66iT - 73T^{2}$$
79 $$1 - 9iT - 79T^{2}$$
83 $$1 - 13.8T + 83T^{2}$$
89 $$1 + 15.5iT - 89T^{2}$$
97 $$1 - 17.3iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$