# Properties

 Label 2-684-228.227-c2-0-56 Degree $2$ Conductor $684$ Sign $-0.451 + 0.892i$ 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 + (−1.90 − 0.613i)2-s + (3.24 + 2.33i)4-s − 6.64i·5-s + 0.470i·7-s + (−4.74 − 6.43i)8-s + (−4.07 + 12.6i)10-s + 10.1·11-s + 3.45i·13-s + (0.288 − 0.896i)14-s + (5.09 + 15.1i)16-s − 14.2i·17-s + (13.6 − 13.2i)19-s + (15.5 − 21.5i)20-s + (−19.3 − 6.24i)22-s + 25.5·23-s + ⋯
 L(s)  = 1 + (−0.951 − 0.306i)2-s + (0.811 + 0.583i)4-s − 1.32i·5-s + 0.0672i·7-s + (−0.593 − 0.804i)8-s + (−0.407 + 1.26i)10-s + 0.924·11-s + 0.265i·13-s + (0.0206 − 0.0640i)14-s + (0.318 + 0.948i)16-s − 0.839i·17-s + (0.717 − 0.696i)19-s + (0.775 − 1.07i)20-s + (−0.880 − 0.283i)22-s + 1.11·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$684$$    =    $$2^{2} \cdot 3^{2} \cdot 19$$ Sign: $-0.451 + 0.892i$ Analytic conductor: $$18.6376$$ Root analytic conductor: $$4.31713$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{684} (683, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 684,\ (\ :1),\ -0.451 + 0.892i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.116764317$$ $$L(\frac12)$$ $$\approx$$ $$1.116764317$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.90 + 0.613i)T$$
3 $$1$$
19 $$1 + (-13.6 + 13.2i)T$$
good5 $$1 + 6.64iT - 25T^{2}$$
7 $$1 - 0.470iT - 49T^{2}$$
11 $$1 - 10.1T + 121T^{2}$$
13 $$1 - 3.45iT - 169T^{2}$$
17 $$1 + 14.2iT - 289T^{2}$$
23 $$1 - 25.5T + 529T^{2}$$
29 $$1 + 14.1T + 841T^{2}$$
31 $$1 + 19.3T + 961T^{2}$$
37 $$1 - 30.0iT - 1.36e3T^{2}$$
41 $$1 - 25.6T + 1.68e3T^{2}$$
43 $$1 + 38.5iT - 1.84e3T^{2}$$
47 $$1 + 16.5T + 2.20e3T^{2}$$
53 $$1 + 41.6T + 2.80e3T^{2}$$
59 $$1 + 31.3iT - 3.48e3T^{2}$$
61 $$1 + 3.53T + 3.72e3T^{2}$$
67 $$1 + 13.9T + 4.48e3T^{2}$$
71 $$1 + 128. iT - 5.04e3T^{2}$$
73 $$1 + 71.6T + 5.32e3T^{2}$$
79 $$1 - 100.T + 6.24e3T^{2}$$
83 $$1 + 58.6T + 6.88e3T^{2}$$
89 $$1 + 108.T + 7.92e3T^{2}$$
97 $$1 + 124. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$