# Properties

 Label 2-147-21.20-c3-0-35 Degree $2$ Conductor $147$ Sign $0.841 - 0.539i$ 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 − 5.37i·2-s + (−2.08 − 4.75i)3-s − 20.8·4-s + 2.78·5-s + (−25.5 + 11.2i)6-s + 69.3i·8-s + (−18.3 + 19.8i)9-s − 14.9i·10-s + 17.5i·11-s + (43.5 + 99.4i)12-s − 47.8i·13-s + (−5.81 − 13.2i)15-s + 205.·16-s − 89.4·17-s + (106. + 98.3i)18-s − 42.2i·19-s + ⋯
 L(s)  = 1 − 1.90i·2-s + (−0.401 − 0.915i)3-s − 2.61·4-s + 0.249·5-s + (−1.74 + 0.762i)6-s + 3.06i·8-s + (−0.677 + 0.735i)9-s − 0.473i·10-s + 0.481i·11-s + (1.04 + 2.39i)12-s − 1.02i·13-s + (−0.100 − 0.228i)15-s + 3.21·16-s − 1.27·17-s + (1.39 + 1.28i)18-s − 0.510i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.841 - 0.539i$ 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: $$(2,\ 147,\ (\ :3/2),\ 0.841 - 0.539i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.280638 + 0.0822421i$$ $$L(\frac12)$$ $$\approx$$ $$0.280638 + 0.0822421i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (2.08 + 4.75i)T$$
7 $$1$$
good2 $$1 + 5.37iT - 8T^{2}$$
5 $$1 - 2.78T + 125T^{2}$$
11 $$1 - 17.5iT - 1.33e3T^{2}$$
13 $$1 + 47.8iT - 2.19e3T^{2}$$
17 $$1 + 89.4T + 4.91e3T^{2}$$
19 $$1 + 42.2iT - 6.85e3T^{2}$$
23 $$1 - 87.6iT - 1.21e4T^{2}$$
29 $$1 + 40.8iT - 2.43e4T^{2}$$
31 $$1 + 95.6iT - 2.97e4T^{2}$$
37 $$1 + 64.5T + 5.06e4T^{2}$$
41 $$1 - 403.T + 6.89e4T^{2}$$
43 $$1 + 230.T + 7.95e4T^{2}$$
47 $$1 + 365.T + 1.03e5T^{2}$$
53 $$1 - 598. iT - 1.48e5T^{2}$$
59 $$1 + 236.T + 2.05e5T^{2}$$
61 $$1 - 430. iT - 2.26e5T^{2}$$
67 $$1 + 428.T + 3.00e5T^{2}$$
71 $$1 + 519. iT - 3.57e5T^{2}$$
73 $$1 + 764. iT - 3.89e5T^{2}$$
79 $$1 + 227.T + 4.93e5T^{2}$$
83 $$1 + 1.13e3T + 5.71e5T^{2}$$
89 $$1 + 1.13e3T + 7.04e5T^{2}$$
97 $$1 + 1.48e3iT - 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

−11.62075359998600227444631017672, −10.95871611261229522338655549458, −9.938578884932182106890663232748, −8.855328310629823930161313334570, −7.63148021232665564677512363088, −5.82007362294415497878478291494, −4.52836093399042612744680753427, −2.82059221615190546445660519975, −1.68469792340236696267768938165, −0.15110160268305911527600071334, 3.97049069948735871500573141960, 4.92710910199368791345363719965, 6.06362768241413878234655862410, 6.79482855669934488055806901825, 8.345852615919036906499218746160, 9.097865800975956824149706916075, 10.01620245073050989584600972854, 11.38583218848288886915512233968, 12.90127031284060428619919588775, 14.04657631613488927995091926184