# Properties

 Label 2-147-7.2-c3-0-14 Degree $2$ Conductor $147$ Sign $0.386 + 0.922i$ 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 + (1.5 − 2.59i)2-s + (1.5 + 2.59i)3-s + (−0.5 − 0.866i)4-s + (9 − 15.5i)5-s + 9·6-s + 21·8-s + (−4.5 + 7.79i)9-s + (−27 − 46.7i)10-s + (18 + 31.1i)11-s + (1.50 − 2.59i)12-s − 34·13-s + 54·15-s + (35.5 − 61.4i)16-s + (−21 − 36.3i)17-s + (13.5 + 23.3i)18-s + (62 − 107. i)19-s + ⋯
 L(s)  = 1 + (0.530 − 0.918i)2-s + (0.288 + 0.499i)3-s + (−0.0625 − 0.108i)4-s + (0.804 − 1.39i)5-s + 0.612·6-s + 0.928·8-s + (−0.166 + 0.288i)9-s + (−0.853 − 1.47i)10-s + (0.493 + 0.854i)11-s + (0.0360 − 0.0625i)12-s − 0.725·13-s + 0.929·15-s + (0.554 − 0.960i)16-s + (−0.299 − 0.518i)17-s + (0.176 + 0.306i)18-s + (0.748 − 1.29i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $0.386 + 0.922i$ Analytic conductor: $$8.67328$$ Root analytic conductor: $$2.94504$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{147} (79, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 147,\ (\ :3/2),\ 0.386 + 0.922i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.42811 - 1.61513i$$ $$L(\frac12)$$ $$\approx$$ $$2.42811 - 1.61513i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.5 - 2.59i)T$$
7 $$1$$
good2 $$1 + (-1.5 + 2.59i)T + (-4 - 6.92i)T^{2}$$
5 $$1 + (-9 + 15.5i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + 34T + 2.19e3T^{2}$$
17 $$1 + (21 + 36.3i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (-62 + 107. i)T + (-3.42e3 - 5.94e3i)T^{2}$$
23 $$1 + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 - 102T + 2.43e4T^{2}$$
31 $$1 + (-80 - 138. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + (199 - 344. i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + 318T + 6.89e4T^{2}$$
43 $$1 + 268T + 7.95e4T^{2}$$
47 $$1 + (120 - 207. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-249 - 431. i)T + (-7.44e4 + 1.28e5i)T^{2}$$
59 $$1 + (-66 - 114. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (199 - 344. i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (46 + 79.6i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 720T + 3.57e5T^{2}$$
73 $$1 + (-251 - 434. i)T + (-1.94e5 + 3.36e5i)T^{2}$$
79 $$1 + (-512 + 886. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 204T + 5.71e5T^{2}$$
89 $$1 + (177 - 306. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 286T + 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

−12.29284372985196174583039302105, −11.74873248680591459369783079130, −10.25129650512381807787937548984, −9.537668631563427251780136069498, −8.570232827982493879575114273638, −7.03892974771903186343672133942, −4.98385409893098249065032521892, −4.62594930458511973997169298755, −2.85178390003630857089305525650, −1.46416920708817542687523475314, 1.93437003949380149975697212137, 3.51970122368449259145734785705, 5.51295160454589611163370816242, 6.37777405805474349100844875411, 7.07073344101855824087974020862, 8.196729264748521278575518083002, 9.830715025281998420737590511703, 10.64483611414046764699936459774, 11.84856313581763872790078226621, 13.30624328202591267409369612700