# Properties

 Label 2-147-21.20-c3-0-7 Degree $2$ Conductor $147$ Sign $-0.875 - 0.483i$ 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 + 1.90i·2-s + (−5.08 + 1.07i)3-s + 4.35·4-s + 1.24·5-s + (−2.05 − 9.70i)6-s + 23.5i·8-s + (24.6 − 10.9i)9-s + 2.38i·10-s + 40.6i·11-s + (−22.1 + 4.69i)12-s + 19.5i·13-s + (−6.34 + 1.34i)15-s − 10.1·16-s − 104.·17-s + (20.9 + 47.0i)18-s + 40.4i·19-s + ⋯
 L(s)  = 1 + 0.674i·2-s + (−0.978 + 0.207i)3-s + 0.544·4-s + 0.111·5-s + (−0.140 − 0.660i)6-s + 1.04i·8-s + (0.913 − 0.406i)9-s + 0.0752i·10-s + 1.11i·11-s + (−0.532 + 0.113i)12-s + 0.418i·13-s + (−0.109 + 0.0231i)15-s − 0.158·16-s − 1.49·17-s + (0.274 + 0.616i)18-s + 0.488i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $-0.875 - 0.483i$ 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: $$(2,\ 147,\ (\ :3/2),\ -0.875 - 0.483i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.279806 + 1.08530i$$ $$L(\frac12)$$ $$\approx$$ $$0.279806 + 1.08530i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (5.08 - 1.07i)T$$
7 $$1$$
good2 $$1 - 1.90iT - 8T^{2}$$
5 $$1 - 1.24T + 125T^{2}$$
11 $$1 - 40.6iT - 1.33e3T^{2}$$
13 $$1 - 19.5iT - 2.19e3T^{2}$$
17 $$1 + 104.T + 4.91e3T^{2}$$
19 $$1 - 40.4iT - 6.85e3T^{2}$$
23 $$1 + 80.4iT - 1.21e4T^{2}$$
29 $$1 - 211. iT - 2.43e4T^{2}$$
31 $$1 - 100. iT - 2.97e4T^{2}$$
37 $$1 + 189.T + 5.06e4T^{2}$$
41 $$1 - 186.T + 6.89e4T^{2}$$
43 $$1 - 158.T + 7.95e4T^{2}$$
47 $$1 - 358.T + 1.03e5T^{2}$$
53 $$1 + 423. iT - 1.48e5T^{2}$$
59 $$1 - 625.T + 2.05e5T^{2}$$
61 $$1 - 807. iT - 2.26e5T^{2}$$
67 $$1 - 298.T + 3.00e5T^{2}$$
71 $$1 + 455. iT - 3.57e5T^{2}$$
73 $$1 - 501. iT - 3.89e5T^{2}$$
79 $$1 + 61.9T + 4.93e5T^{2}$$
83 $$1 - 73.1T + 5.71e5T^{2}$$
89 $$1 - 114.T + 7.04e5T^{2}$$
97 $$1 + 1.41e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$