Properties

Label 2-147-21.20-c3-0-24
Degree $2$
Conductor $147$
Sign $0.603 + 0.797i$
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  − 1.90i·2-s + (5.08 + 1.07i)3-s + 4.35·4-s − 1.24·5-s + (2.05 − 9.70i)6-s − 23.5i·8-s + (24.6 + 10.9i)9-s + 2.38i·10-s − 40.6i·11-s + (22.1 + 4.69i)12-s + 19.5i·13-s + (−6.34 − 1.34i)15-s − 10.1·16-s + 104.·17-s + (20.9 − 47.0i)18-s + 40.4i·19-s + ⋯
L(s)  = 1  − 0.674i·2-s + (0.978 + 0.207i)3-s + 0.544·4-s − 0.111·5-s + (0.140 − 0.660i)6-s − 1.04i·8-s + (0.913 + 0.406i)9-s + 0.0752i·10-s − 1.11i·11-s + (0.532 + 0.113i)12-s + 0.418i·13-s + (−0.109 − 0.0231i)15-s − 0.158·16-s + 1.49·17-s + (0.274 − 0.616i)18-s + 0.488i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.603 + 0.797i$
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: \((2,\ 147,\ (\ :3/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.41812 - 1.20229i\)
\(L(\frac12)\) \(\approx\) \(2.41812 - 1.20229i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.08 - 1.07i)T \)
7 \( 1 \)
good2 \( 1 + 1.90iT - 8T^{2} \)
5 \( 1 + 1.24T + 125T^{2} \)
11 \( 1 + 40.6iT - 1.33e3T^{2} \)
13 \( 1 - 19.5iT - 2.19e3T^{2} \)
17 \( 1 - 104.T + 4.91e3T^{2} \)
19 \( 1 - 40.4iT - 6.85e3T^{2} \)
23 \( 1 - 80.4iT - 1.21e4T^{2} \)
29 \( 1 + 211. iT - 2.43e4T^{2} \)
31 \( 1 - 100. iT - 2.97e4T^{2} \)
37 \( 1 + 189.T + 5.06e4T^{2} \)
41 \( 1 + 186.T + 6.89e4T^{2} \)
43 \( 1 - 158.T + 7.95e4T^{2} \)
47 \( 1 + 358.T + 1.03e5T^{2} \)
53 \( 1 - 423. iT - 1.48e5T^{2} \)
59 \( 1 + 625.T + 2.05e5T^{2} \)
61 \( 1 - 807. iT - 2.26e5T^{2} \)
67 \( 1 - 298.T + 3.00e5T^{2} \)
71 \( 1 - 455. iT - 3.57e5T^{2} \)
73 \( 1 - 501. iT - 3.89e5T^{2} \)
79 \( 1 + 61.9T + 4.93e5T^{2} \)
83 \( 1 + 73.1T + 5.71e5T^{2} \)
89 \( 1 + 114.T + 7.04e5T^{2} \)
97 \( 1 + 1.41e3iT - 9.12e5T^{2} \)
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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

−12.29614800007504265900496382341, −11.46856450496218195207552932519, −10.31943302629787029909550132683, −9.582031406829580329498950410426, −8.272766600901585193667570808363, −7.34987610288393528786273077513, −5.86105061085640865456517924124, −3.88002577213761281613107193604, −3.00362468584593911877343926230, −1.48429823417206950231400203214, 1.86676518890572452856189722313, 3.29578082019675778562190280430, 5.04630227575658032713222513779, 6.57666140000918656083693043207, 7.51234853663699067284623015225, 8.190851573757592475578351149871, 9.515831868297571701338804487112, 10.52475200297306339456175832803, 11.98987923491289114058636415921, 12.74851475524238836445542292774

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