Properties

Degree $2$
Conductor $147$
Sign $0.386 + 0.922i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

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: $0.386 + 0.922i$
Motivic weight: \(3\)
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

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