# Properties

 Label 2-147-21.20-c3-0-2 Degree $2$ Conductor $147$ Sign $0.969 + 0.243i$ Analytic cond. $8.67328$ Root an. cond. $2.94504$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 4.84i·2-s + (−5.11 + 0.905i)3-s − 15.4·4-s − 17.3·5-s + (4.38 + 24.7i)6-s + 36.2i·8-s + (25.3 − 9.26i)9-s + 83.8i·10-s − 27.8i·11-s + (79.1 − 14.0i)12-s − 16.3i·13-s + (88.5 − 15.6i)15-s + 51.6·16-s + 40.8·17-s + (−44.8 − 122. i)18-s + 68.1i·19-s + ⋯
 L(s)  = 1 − 1.71i·2-s + (−0.984 + 0.174i)3-s − 1.93·4-s − 1.54·5-s + (0.298 + 1.68i)6-s + 1.59i·8-s + (0.939 − 0.343i)9-s + 2.65i·10-s − 0.763i·11-s + (1.90 − 0.336i)12-s − 0.347i·13-s + (1.52 − 0.269i)15-s + 0.806·16-s + 0.582·17-s + (−0.587 − 1.60i)18-s + 0.823i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.969 + 0.243i$ Analytic conductor: $$8.67328$$ Root analytic conductor: $$2.94504$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{147} (146, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :3/2),\ 0.969 + 0.243i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.324731 - 0.0402115i$$ $$L(\frac12)$$ $$\approx$$ $$0.324731 - 0.0402115i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (5.11 - 0.905i)T$$
7 $$1$$
good2 $$1 + 4.84iT - 8T^{2}$$
5 $$1 + 17.3T + 125T^{2}$$
11 $$1 + 27.8iT - 1.33e3T^{2}$$
13 $$1 + 16.3iT - 2.19e3T^{2}$$
17 $$1 - 40.8T + 4.91e3T^{2}$$
19 $$1 - 68.1iT - 6.85e3T^{2}$$
23 $$1 - 71.7iT - 1.21e4T^{2}$$
29 $$1 - 216. iT - 2.43e4T^{2}$$
31 $$1 + 157. iT - 2.97e4T^{2}$$
37 $$1 + 348.T + 5.06e4T^{2}$$
41 $$1 - 153.T + 6.89e4T^{2}$$
43 $$1 - 427.T + 7.95e4T^{2}$$
47 $$1 - 16.8T + 1.03e5T^{2}$$
53 $$1 + 192. iT - 1.48e5T^{2}$$
59 $$1 + 287.T + 2.05e5T^{2}$$
61 $$1 - 224. iT - 2.26e5T^{2}$$
67 $$1 + 172.T + 3.00e5T^{2}$$
71 $$1 - 1.06e3iT - 3.57e5T^{2}$$
73 $$1 - 1.07e3iT - 3.89e5T^{2}$$
79 $$1 - 52.0T + 4.93e5T^{2}$$
83 $$1 + 1.02e3T + 5.71e5T^{2}$$
89 $$1 - 668.T + 7.04e5T^{2}$$
97 $$1 + 1.35e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$