# Properties

 Label 2-147-21.20-c3-0-19 Degree $2$ Conductor $147$ Sign $0.945 + 0.326i$ Analytic cond. $8.67328$ Root an. cond. $2.94504$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2.58i·2-s + (2.60 + 4.49i)3-s + 1.30·4-s + 16.1·5-s + (11.6 − 6.73i)6-s − 24.0i·8-s + (−13.4 + 23.4i)9-s − 41.7i·10-s + 35.5i·11-s + (3.39 + 5.87i)12-s + 7.40i·13-s + (41.9 + 72.4i)15-s − 51.8·16-s + 28.9·17-s + (60.5 + 34.8i)18-s − 35.1i·19-s + ⋯
 L(s)  = 1 − 0.914i·2-s + (0.500 + 0.865i)3-s + 0.163·4-s + 1.44·5-s + (0.791 − 0.458i)6-s − 1.06i·8-s + (−0.498 + 0.867i)9-s − 1.31i·10-s + 0.975i·11-s + (0.0817 + 0.141i)12-s + 0.158i·13-s + (0.722 + 1.24i)15-s − 0.810·16-s + 0.412·17-s + (0.793 + 0.455i)18-s − 0.424i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.945 + 0.326i$ 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: $$(2,\ 147,\ (\ :3/2),\ 0.945 + 0.326i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.64372 - 0.443578i$$ $$L(\frac12)$$ $$\approx$$ $$2.64372 - 0.443578i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-2.60 - 4.49i)T$$
7 $$1$$
good2 $$1 + 2.58iT - 8T^{2}$$
5 $$1 - 16.1T + 125T^{2}$$
11 $$1 - 35.5iT - 1.33e3T^{2}$$
13 $$1 - 7.40iT - 2.19e3T^{2}$$
17 $$1 - 28.9T + 4.91e3T^{2}$$
19 $$1 + 35.1iT - 6.85e3T^{2}$$
23 $$1 + 55.4iT - 1.21e4T^{2}$$
29 $$1 - 68.1iT - 2.43e4T^{2}$$
31 $$1 + 178. iT - 2.97e4T^{2}$$
37 $$1 + 233.T + 5.06e4T^{2}$$
41 $$1 - 370.T + 6.89e4T^{2}$$
43 $$1 + 187.T + 7.95e4T^{2}$$
47 $$1 - 174.T + 1.03e5T^{2}$$
53 $$1 + 272. iT - 1.48e5T^{2}$$
59 $$1 - 96.8T + 2.05e5T^{2}$$
61 $$1 - 385. iT - 2.26e5T^{2}$$
67 $$1 + 1.01e3T + 3.00e5T^{2}$$
71 $$1 - 125. iT - 3.57e5T^{2}$$
73 $$1 - 225. iT - 3.89e5T^{2}$$
79 $$1 + 1.06e3T + 4.93e5T^{2}$$
83 $$1 + 601.T + 5.71e5T^{2}$$
89 $$1 + 1.50e3T + 7.04e5T^{2}$$
97 $$1 + 327. iT - 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−12.55603583807526259835488090259, −11.29377468331514643024833131513, −10.23760081333176737175936886535, −9.837251476749226921511216936168, −8.918029035532858419244590502888, −7.16947432253756454767930433972, −5.75676351430161776878071920748, −4.36511763763900482305210713198, −2.81069709100425223123977288489, −1.83411433169953708913262486501, 1.59333761152944019449775745938, 2.93509229163369546656775984105, 5.57956309014487267696872622332, 6.12431268211630971668610239460, 7.19671870935127085798065890733, 8.282663190684389873484206717699, 9.197367244292218906594881334008, 10.51865256364668570507047281425, 11.79855571523008641232817486381, 12.97789416791434908033728179076