Properties

Label 2-147-1.1-c3-0-2
Degree $2$
Conductor $147$
Sign $1$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 3·3-s − 7.82·4-s + 0.100·5-s + 1.24·6-s + 6.55·8-s + 9·9-s − 0.0416·10-s − 43.9·11-s + 23.4·12-s + 16.6·13-s − 0.301·15-s + 59.9·16-s + 121.·17-s − 3.72·18-s + 127.·19-s − 0.786·20-s + 18.2·22-s + 53.5·23-s − 19.6·24-s − 124.·25-s − 6.89·26-s − 27·27-s + 235.·29-s + 0.124·30-s + 18.7·31-s − 77.2·32-s + ⋯
L(s)  = 1  − 0.146·2-s − 0.577·3-s − 0.978·4-s + 0.00898·5-s + 0.0845·6-s + 0.289·8-s + 0.333·9-s − 0.00131·10-s − 1.20·11-s + 0.564·12-s + 0.355·13-s − 0.00519·15-s + 0.936·16-s + 1.73·17-s − 0.0488·18-s + 1.53·19-s − 0.00879·20-s + 0.176·22-s + 0.485·23-s − 0.167·24-s − 0.999·25-s − 0.0520·26-s − 0.192·27-s + 1.50·29-s + 0.000760·30-s + 0.108·31-s − 0.426·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9569825915\)
\(L(\frac12)\) \(\approx\) \(0.9569825915\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
7 \( 1 \)
good2 \( 1 + 0.414T + 8T^{2} \)
5 \( 1 - 0.100T + 125T^{2} \)
11 \( 1 + 43.9T + 1.33e3T^{2} \)
13 \( 1 - 16.6T + 2.19e3T^{2} \)
17 \( 1 - 121.T + 4.91e3T^{2} \)
19 \( 1 - 127.T + 6.85e3T^{2} \)
23 \( 1 - 53.5T + 1.21e4T^{2} \)
29 \( 1 - 235.T + 2.43e4T^{2} \)
31 \( 1 - 18.7T + 2.97e4T^{2} \)
37 \( 1 + 191.T + 5.06e4T^{2} \)
41 \( 1 - 319.T + 6.89e4T^{2} \)
43 \( 1 + 218.T + 7.95e4T^{2} \)
47 \( 1 + 401.T + 1.03e5T^{2} \)
53 \( 1 - 643.T + 1.48e5T^{2} \)
59 \( 1 - 11.6T + 2.05e5T^{2} \)
61 \( 1 - 12.2T + 2.26e5T^{2} \)
67 \( 1 - 669.T + 3.00e5T^{2} \)
71 \( 1 - 822.T + 3.57e5T^{2} \)
73 \( 1 + 515.T + 3.89e5T^{2} \)
79 \( 1 + 805.T + 4.93e5T^{2} \)
83 \( 1 - 394.T + 5.71e5T^{2} \)
89 \( 1 + 673.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3T + 9.12e5T^{2} \)
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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

−12.60153794048880854982947016504, −11.66225681106561843424255459710, −10.28714456508783178053882430454, −9.768600631814430345034724938467, −8.331574874294928749970975040208, −7.46077453457776503172401273963, −5.69489440829089774115211728652, −4.97533458659597421169175463349, −3.37102006277055577135782932611, −0.877718337794786979014334370412, 0.877718337794786979014334370412, 3.37102006277055577135782932611, 4.97533458659597421169175463349, 5.69489440829089774115211728652, 7.46077453457776503172401273963, 8.331574874294928749970975040208, 9.768600631814430345034724938467, 10.28714456508783178053882430454, 11.66225681106561843424255459710, 12.60153794048880854982947016504

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