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Properties

Label 2-147-21.20-c3-0-10
Degree $2$
Conductor $147$
Sign $-0.361 - 0.932i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.222i·2-s + (3.69 + 3.65i)3-s + 7.95·4-s − 18.7·5-s + (−0.814 + 0.823i)6-s + 3.55i·8-s + (0.292 + 26.9i)9-s − 4.16i·10-s + 45.8i·11-s + (29.3 + 29.0i)12-s + 59.3i·13-s + (−69.0 − 68.3i)15-s + 62.8·16-s − 28.9·17-s + (−6.01 + 0.0652i)18-s − 38.9i·19-s + ⋯
L(s)  = 1  + 0.0788i·2-s + (0.710 + 0.703i)3-s + 0.993·4-s − 1.67·5-s + (−0.0554 + 0.0560i)6-s + 0.157i·8-s + (0.0108 + 0.999i)9-s − 0.131i·10-s + 1.25i·11-s + (0.706 + 0.698i)12-s + 1.26i·13-s + (−1.18 − 1.17i)15-s + 0.981·16-s − 0.412·17-s + (−0.0788 + 0.000854i)18-s − 0.470i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.361 - 0.932i$
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: \((2,\ 147,\ (\ :3/2),\ -0.361 - 0.932i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.990072 + 1.44493i\)
\(L(\frac12)\) \(\approx\) \(0.990072 + 1.44493i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.69 - 3.65i)T \)
7 \( 1 \)
good2 \( 1 - 0.222iT - 8T^{2} \)
5 \( 1 + 18.7T + 125T^{2} \)
11 \( 1 - 45.8iT - 1.33e3T^{2} \)
13 \( 1 - 59.3iT - 2.19e3T^{2} \)
17 \( 1 + 28.9T + 4.91e3T^{2} \)
19 \( 1 + 38.9iT - 6.85e3T^{2} \)
23 \( 1 + 1.79iT - 1.21e4T^{2} \)
29 \( 1 + 148. iT - 2.43e4T^{2} \)
31 \( 1 - 104. iT - 2.97e4T^{2} \)
37 \( 1 + 24.1T + 5.06e4T^{2} \)
41 \( 1 - 254.T + 6.89e4T^{2} \)
43 \( 1 + 59.6T + 7.95e4T^{2} \)
47 \( 1 - 262.T + 1.03e5T^{2} \)
53 \( 1 + 381. iT - 1.48e5T^{2} \)
59 \( 1 - 371.T + 2.05e5T^{2} \)
61 \( 1 + 696. iT - 2.26e5T^{2} \)
67 \( 1 - 434.T + 3.00e5T^{2} \)
71 \( 1 - 824. iT - 3.57e5T^{2} \)
73 \( 1 + 0.395iT - 3.89e5T^{2} \)
79 \( 1 + 731.T + 4.93e5T^{2} \)
83 \( 1 - 586.T + 5.71e5T^{2} \)
89 \( 1 + 440.T + 7.04e5T^{2} \)
97 \( 1 - 634. iT - 9.12e5T^{2} \)
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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

−12.67747177761381026997304544269, −11.67154684176650566960050838979, −11.07883415987127876781683729218, −9.847556046709533244223100579163, −8.610927673052930148195149105176, −7.57562299607866707020389288322, −6.87234722318976349298217975553, −4.68926469758412721593860527075, −3.78292621323443505123916238120, −2.28234404520363551408239034768, 0.78204504095524429687381801182, 2.87222500847528459737557773759, 3.71896471581133342659389058417, 5.94969654243712176415178313103, 7.25237155252125766760573665202, 7.908387236462060946006921725998, 8.702608164356903498078418098057, 10.57999284172132471727293534021, 11.41808129419852355986108670710, 12.22697438112230728847662985436

Graph of the $Z$-function along the critical line