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Properties

Label 2-147-7.3-c6-0-9
Degree $2$
Conductor $147$
Sign $0.553 - 0.832i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.76 − 4.78i)2-s + (13.5 + 7.79i)3-s + (16.7 − 28.9i)4-s + (57.9 − 33.4i)5-s − 86.1i·6-s − 538.·8-s + (121.5 + 210. i)9-s + (−320. − 185. i)10-s + (−862. + 1.49e3i)11-s + (451. − 260. i)12-s + 2.80e3i·13-s + 1.04e3·15-s + (418. + 724. i)16-s + (5.32e3 + 3.07e3i)17-s + (671. − 1.16e3i)18-s + (−7.73e3 + 4.46e3i)19-s + ⋯
L(s)  = 1  + (−0.345 − 0.598i)2-s + (0.5 + 0.288i)3-s + (0.261 − 0.452i)4-s + (0.463 − 0.267i)5-s − 0.398i·6-s − 1.05·8-s + (0.166 + 0.288i)9-s + (−0.320 − 0.185i)10-s + (−0.648 + 1.12i)11-s + (0.261 − 0.150i)12-s + 1.27i·13-s + 0.309·15-s + (0.102 + 0.176i)16-s + (1.08 + 0.625i)17-s + (0.115 − 0.199i)18-s + (−1.12 + 0.651i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.476653637\)
\(L(\frac12)\) \(\approx\) \(1.476653637\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.5 - 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (2.76 + 4.78i)T + (-32 + 55.4i)T^{2} \)
5 \( 1 + (-57.9 + 33.4i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (862. - 1.49e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 2.80e3iT - 4.82e6T^{2} \)
17 \( 1 + (-5.32e3 - 3.07e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (7.73e3 - 4.46e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (4.95e3 + 8.57e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 1.36e4T + 5.94e8T^{2} \)
31 \( 1 + (-2.18e4 - 1.26e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (1.13e4 + 1.96e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 3.78e4iT - 4.75e9T^{2} \)
43 \( 1 - 7.36e4T + 6.32e9T^{2} \)
47 \( 1 + (1.20e5 - 6.95e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-8.25e3 + 1.43e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-6.19e4 - 3.57e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-2.28e5 + 1.31e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (2.74e5 - 4.75e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 4.65e5T + 1.28e11T^{2} \)
73 \( 1 + (-2.89e5 - 1.66e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.05e5 - 3.55e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 7.19e4iT - 3.26e11T^{2} \)
89 \( 1 + (-1.41e5 + 8.16e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 6.51e5iT - 8.32e11T^{2} \)
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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

−12.08726030189835464860960232456, −10.78698459659294127162649277669, −9.988160275176187633199376589886, −9.349685827942183795794275962528, −8.169217626317966988219790037341, −6.71752340432493837221259425335, −5.46377761705833870894080259561, −4.05089819683749706283587237300, −2.34150826849915547483604245286, −1.56905353453150460977820837023, 0.43292272083223145598671803054, 2.50878589491999547156869766047, 3.38203522458450233917077036145, 5.54916402589112211554389719575, 6.48887053831463964436727045198, 7.81374446039876661815943549865, 8.240112742786527922370990958066, 9.510887845741560470607903398154, 10.64537394154840904461266493934, 11.84755388189070855812519914356

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