# Properties

 Label 2-147-21.20-c3-0-23 Degree $2$ Conductor $147$ Sign $-0.967 + 0.253i$ 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 − 4.54i·2-s + (−2.93 + 4.28i)3-s − 12.6·4-s + 11.6·5-s + (19.4 + 13.3i)6-s + 21.1i·8-s + (−9.72 − 25.1i)9-s − 52.7i·10-s − 17.9i·11-s + (37.2 − 54.2i)12-s − 62.4i·13-s + (−34.1 + 49.7i)15-s − 4.97·16-s − 21.4·17-s + (−114. + 44.1i)18-s + 10.9i·19-s + ⋯
 L(s)  = 1 − 1.60i·2-s + (−0.565 + 0.824i)3-s − 1.58·4-s + 1.03·5-s + (1.32 + 0.909i)6-s + 0.936i·8-s + (−0.360 − 0.932i)9-s − 1.66i·10-s − 0.491i·11-s + (0.895 − 1.30i)12-s − 1.33i·13-s + (−0.587 + 0.855i)15-s − 0.0777·16-s − 0.305·17-s + (−1.49 + 0.578i)18-s + 0.132i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$147$$    =    $$3 \cdot 7^{2}$$ Sign: $-0.967 + 0.253i$ 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: $$(2,\ 147,\ (\ :3/2),\ -0.967 + 0.253i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.151205 - 1.17570i$$ $$L(\frac12)$$ $$\approx$$ $$0.151205 - 1.17570i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (2.93 - 4.28i)T$$
7 $$1$$
good2 $$1 + 4.54iT - 8T^{2}$$
5 $$1 - 11.6T + 125T^{2}$$
11 $$1 + 17.9iT - 1.33e3T^{2}$$
13 $$1 + 62.4iT - 2.19e3T^{2}$$
17 $$1 + 21.4T + 4.91e3T^{2}$$
19 $$1 - 10.9iT - 6.85e3T^{2}$$
23 $$1 + 69.0iT - 1.21e4T^{2}$$
29 $$1 + 265. iT - 2.43e4T^{2}$$
31 $$1 + 10.2iT - 2.97e4T^{2}$$
37 $$1 - 41.6T + 5.06e4T^{2}$$
41 $$1 + 31.0T + 6.89e4T^{2}$$
43 $$1 + 224.T + 7.95e4T^{2}$$
47 $$1 - 163.T + 1.03e5T^{2}$$
53 $$1 + 527. iT - 1.48e5T^{2}$$
59 $$1 - 411.T + 2.05e5T^{2}$$
61 $$1 - 258. iT - 2.26e5T^{2}$$
67 $$1 - 323.T + 3.00e5T^{2}$$
71 $$1 - 45.4iT - 3.57e5T^{2}$$
73 $$1 - 562. iT - 3.89e5T^{2}$$
79 $$1 - 289.T + 4.93e5T^{2}$$
83 $$1 - 448.T + 5.71e5T^{2}$$
89 $$1 - 561.T + 7.04e5T^{2}$$
97 $$1 - 214. 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

−11.87940002681837916773751380510, −10.99749672237722726062820724094, −10.16340048790787467864080963877, −9.698112972271994755018794711799, −8.500113818341542987149945344452, −6.20885494678018047826560603048, −5.14563201964165086096587903148, −3.77035271051439953526444996400, −2.47777060800753002690735939082, −0.62027723856959913856128302700, 1.86103293961987726992924415927, 4.77021985189456819640541018994, 5.75218613031618407075166483776, 6.66554560972888642046752556359, 7.29937913766839032608454239276, 8.641670817413848607506635261987, 9.599943640028215080378400228488, 11.08309288038581043537897631844, 12.34158112892268832409587813989, 13.51156201481918138024621182113