# Properties

 Label 2-684-3.2-c2-0-1 Degree $2$ Conductor $684$ Sign $-0.577 - 0.816i$ Analytic cond. $18.6376$ Root an. cond. $4.31713$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.29i·5-s − 2.46·7-s − 10.4i·11-s + 2.93·13-s + 27.8i·17-s − 4.35·19-s + 24.3i·23-s + 14.1·25-s + 7.80i·29-s − 17.4·31-s − 8.11i·35-s − 48.3·37-s + 51.2i·41-s − 82.7·43-s − 22.2i·47-s + ⋯
 L(s)  = 1 + 0.659i·5-s − 0.351·7-s − 0.946i·11-s + 0.225·13-s + 1.63i·17-s − 0.229·19-s + 1.05i·23-s + 0.565·25-s + 0.269i·29-s − 0.562·31-s − 0.231i·35-s − 1.30·37-s + 1.24i·41-s − 1.92·43-s − 0.472i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$684$$    =    $$2^{2} \cdot 3^{2} \cdot 19$$ Sign: $-0.577 - 0.816i$ Analytic conductor: $$18.6376$$ Root analytic conductor: $$4.31713$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{684} (305, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 684,\ (\ :1),\ -0.577 - 0.816i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.028223706$$ $$L(\frac12)$$ $$\approx$$ $$1.028223706$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
19 $$1 + 4.35T$$
good5 $$1 - 3.29iT - 25T^{2}$$
7 $$1 + 2.46T + 49T^{2}$$
11 $$1 + 10.4iT - 121T^{2}$$
13 $$1 - 2.93T + 169T^{2}$$
17 $$1 - 27.8iT - 289T^{2}$$
23 $$1 - 24.3iT - 529T^{2}$$
29 $$1 - 7.80iT - 841T^{2}$$
31 $$1 + 17.4T + 961T^{2}$$
37 $$1 + 48.3T + 1.36e3T^{2}$$
41 $$1 - 51.2iT - 1.68e3T^{2}$$
43 $$1 + 82.7T + 1.84e3T^{2}$$
47 $$1 + 22.2iT - 2.20e3T^{2}$$
53 $$1 - 16.6iT - 2.80e3T^{2}$$
59 $$1 - 49.5iT - 3.48e3T^{2}$$
61 $$1 - 16.9T + 3.72e3T^{2}$$
67 $$1 + 1.05T + 4.48e3T^{2}$$
71 $$1 - 79.0iT - 5.04e3T^{2}$$
73 $$1 + 53.4T + 5.32e3T^{2}$$
79 $$1 - 137.T + 6.24e3T^{2}$$
83 $$1 + 4.56iT - 6.88e3T^{2}$$
89 $$1 - 120. iT - 7.92e3T^{2}$$
97 $$1 - 13.4T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$