# Properties

 Label 2-189-21.20-c1-0-8 Degree $2$ Conductor $189$ Sign $-0.654 + 0.755i$ Analytic cond. $1.50917$ Root an. cond. $1.22848$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.23i·2-s − 3.00·4-s + 3.87·5-s + (−2 − 1.73i)7-s + 2.23i·8-s − 8.66i·10-s − 2.23i·11-s + 3.46i·13-s + (−3.87 + 4.47i)14-s − 0.999·16-s + 5.19i·19-s − 11.6·20-s − 5.00·22-s − 2.23i·23-s + 10.0·25-s + 7.74·26-s + ⋯
 L(s)  = 1 − 1.58i·2-s − 1.50·4-s + 1.73·5-s + (−0.755 − 0.654i)7-s + 0.790i·8-s − 2.73i·10-s − 0.674i·11-s + 0.960i·13-s + (−1.03 + 1.19i)14-s − 0.249·16-s + 1.19i·19-s − 2.59·20-s − 1.06·22-s − 0.466i·23-s + 2.00·25-s + 1.51·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{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: $$189$$    =    $$3^{3} \cdot 7$$ Sign: $-0.654 + 0.755i$ Analytic conductor: $$1.50917$$ Root analytic conductor: $$1.22848$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{189} (188, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 189,\ (\ :1/2),\ -0.654 + 0.755i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.547839 - 1.19916i$$ $$L(\frac12)$$ $$\approx$$ $$0.547839 - 1.19916i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (2 + 1.73i)T$$
good2 $$1 + 2.23iT - 2T^{2}$$
5 $$1 - 3.87T + 5T^{2}$$
11 $$1 + 2.23iT - 11T^{2}$$
13 $$1 - 3.46iT - 13T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 - 5.19iT - 19T^{2}$$
23 $$1 + 2.23iT - 23T^{2}$$
29 $$1 - 4.47iT - 29T^{2}$$
31 $$1 + 1.73iT - 31T^{2}$$
37 $$1 + T + 37T^{2}$$
41 $$1 - 3.87T + 41T^{2}$$
43 $$1 - 2T + 43T^{2}$$
47 $$1 - 7.74T + 47T^{2}$$
53 $$1 + 8.94iT - 53T^{2}$$
59 $$1 + 7.74T + 59T^{2}$$
61 $$1 - 6.92iT - 61T^{2}$$
67 $$1 + 10T + 67T^{2}$$
71 $$1 - 11.1iT - 71T^{2}$$
73 $$1 - 10.3iT - 73T^{2}$$
79 $$1 - 2T + 79T^{2}$$
83 $$1 - 7.74T + 83T^{2}$$
89 $$1 + 11.6T + 89T^{2}$$
97 $$1 + 13.8iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$