Properties

Label 2-147-1.1-c3-0-7
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  + 2.41·2-s − 3·3-s − 2.17·4-s + 19.8·5-s − 7.24·6-s − 24.5·8-s + 9·9-s + 48.0·10-s + 23.9·11-s + 6.51·12-s + 87.3·13-s − 59.6·15-s − 41.9·16-s − 5.63·17-s + 21.7·18-s + 64.8·19-s − 43.2·20-s + 57.7·22-s − 25.5·23-s + 73.6·24-s + 270.·25-s + 210.·26-s − 27·27-s + 60.3·29-s − 144.·30-s − 122.·31-s + 95.2·32-s + ⋯
L(s)  = 1  + 0.853·2-s − 0.577·3-s − 0.271·4-s + 1.77·5-s − 0.492·6-s − 1.08·8-s + 0.333·9-s + 1.51·10-s + 0.656·11-s + 0.156·12-s + 1.86·13-s − 1.02·15-s − 0.654·16-s − 0.0804·17-s + 0.284·18-s + 0.783·19-s − 0.483·20-s + 0.560·22-s − 0.232·23-s + 0.626·24-s + 2.16·25-s + 1.59·26-s − 0.192·27-s + 0.386·29-s − 0.877·30-s − 0.710·31-s + 0.526·32-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

\(L(2)\) \(\approx\) \(2.570168250\)
\(L(\frac12)\) \(\approx\) \(2.570168250\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.41T + 8T^{2} \)
5 \( 1 - 19.8T + 125T^{2} \)
11 \( 1 - 23.9T + 1.33e3T^{2} \)
13 \( 1 - 87.3T + 2.19e3T^{2} \)
17 \( 1 + 5.63T + 4.91e3T^{2} \)
19 \( 1 - 64.8T + 6.85e3T^{2} \)
23 \( 1 + 25.5T + 1.21e4T^{2} \)
29 \( 1 - 60.3T + 2.43e4T^{2} \)
31 \( 1 + 122.T + 2.97e4T^{2} \)
37 \( 1 + 56.1T + 5.06e4T^{2} \)
41 \( 1 + 299.T + 6.89e4T^{2} \)
43 \( 1 + 501.T + 7.95e4T^{2} \)
47 \( 1 - 305.T + 1.03e5T^{2} \)
53 \( 1 + 375.T + 1.48e5T^{2} \)
59 \( 1 + 627.T + 2.05e5T^{2} \)
61 \( 1 - 3.75T + 2.26e5T^{2} \)
67 \( 1 + 813.T + 3.00e5T^{2} \)
71 \( 1 - 165.T + 3.57e5T^{2} \)
73 \( 1 - 619.T + 3.89e5T^{2} \)
79 \( 1 + 138.T + 4.93e5T^{2} \)
83 \( 1 - 621.T + 5.71e5T^{2} \)
89 \( 1 - 285.T + 7.04e5T^{2} \)
97 \( 1 + 603.T + 9.12e5T^{2} \)
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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

−12.89013581185121024513387114812, −11.77658331354819561191606962764, −10.59945077944475582313314238480, −9.532896659771571726199669550105, −8.725739071186004366609981559003, −6.50192839675392189190964801012, −5.93476750785125016658678600706, −4.98876506615176824019409913789, −3.45661462245391568881073052802, −1.45458489799164087560544367544, 1.45458489799164087560544367544, 3.45661462245391568881073052802, 4.98876506615176824019409913789, 5.93476750785125016658678600706, 6.50192839675392189190964801012, 8.725739071186004366609981559003, 9.532896659771571726199669550105, 10.59945077944475582313314238480, 11.77658331354819561191606962764, 12.89013581185121024513387114812

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