Properties

Label 2-147-1.1-c5-0-33
Degree $2$
Conductor $147$
Sign $-1$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.38·2-s + 9·3-s + 8.83·4-s − 38.7·5-s + 57.5·6-s − 148.·8-s + 81·9-s − 247.·10-s − 576.·11-s + 79.4·12-s − 391.·13-s − 348.·15-s − 1.22e3·16-s + 1.32e3·17-s + 517.·18-s − 942.·19-s − 341.·20-s − 3.68e3·22-s − 1.63e3·23-s − 1.33e3·24-s − 1.62e3·25-s − 2.50e3·26-s + 729·27-s − 1.46e3·29-s − 2.22e3·30-s + 3.91e3·31-s − 3.11e3·32-s + ⋯
L(s)  = 1  + 1.12·2-s + 0.577·3-s + 0.275·4-s − 0.692·5-s + 0.652·6-s − 0.817·8-s + 0.333·9-s − 0.782·10-s − 1.43·11-s + 0.159·12-s − 0.642·13-s − 0.399·15-s − 1.19·16-s + 1.11·17-s + 0.376·18-s − 0.598·19-s − 0.191·20-s − 1.62·22-s − 0.643·23-s − 0.472·24-s − 0.520·25-s − 0.725·26-s + 0.192·27-s − 0.323·29-s − 0.451·30-s + 0.731·31-s − 0.537·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
7 \( 1 \)
good2 \( 1 - 6.38T + 32T^{2} \)
5 \( 1 + 38.7T + 3.12e3T^{2} \)
11 \( 1 + 576.T + 1.61e5T^{2} \)
13 \( 1 + 391.T + 3.71e5T^{2} \)
17 \( 1 - 1.32e3T + 1.41e6T^{2} \)
19 \( 1 + 942.T + 2.47e6T^{2} \)
23 \( 1 + 1.63e3T + 6.43e6T^{2} \)
29 \( 1 + 1.46e3T + 2.05e7T^{2} \)
31 \( 1 - 3.91e3T + 2.86e7T^{2} \)
37 \( 1 + 1.63e4T + 6.93e7T^{2} \)
41 \( 1 - 1.31e4T + 1.15e8T^{2} \)
43 \( 1 - 1.47e4T + 1.47e8T^{2} \)
47 \( 1 - 6.81e3T + 2.29e8T^{2} \)
53 \( 1 + 2.01e3T + 4.18e8T^{2} \)
59 \( 1 + 5.14e4T + 7.14e8T^{2} \)
61 \( 1 + 4.10e4T + 8.44e8T^{2} \)
67 \( 1 - 5.05e4T + 1.35e9T^{2} \)
71 \( 1 - 3.99e4T + 1.80e9T^{2} \)
73 \( 1 - 5.56e4T + 2.07e9T^{2} \)
79 \( 1 + 6.31e4T + 3.07e9T^{2} \)
83 \( 1 + 4.55e4T + 3.93e9T^{2} \)
89 \( 1 + 1.56e4T + 5.58e9T^{2} \)
97 \( 1 + 3.12e3T + 8.58e9T^{2} \)
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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

−12.18531646024113554861615028485, −10.75025679854187434642862371755, −9.597832961713370677363959296157, −8.252508069274632845332126271264, −7.43894621286793819805019524320, −5.80905837720830070576998366880, −4.71988752942268134871114721622, −3.61859773359340275618543275722, −2.51122278055465735574161802883, 0, 2.51122278055465735574161802883, 3.61859773359340275618543275722, 4.71988752942268134871114721622, 5.80905837720830070576998366880, 7.43894621286793819805019524320, 8.252508069274632845332126271264, 9.597832961713370677363959296157, 10.75025679854187434642862371755, 12.18531646024113554861615028485

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