Properties

Label 2-147-1.1-c5-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.44·2-s − 9·3-s + 39.3·4-s + 36·5-s + 76.0·6-s − 61.9·8-s + 81·9-s − 304.·10-s + 295.·11-s − 354.·12-s + 1.14e3·13-s − 324·15-s − 735.·16-s − 1.03e3·17-s − 684.·18-s − 2.10e3·19-s + 1.41e3·20-s − 2.49e3·22-s − 640.·23-s + 557.·24-s − 1.82e3·25-s − 9.69e3·26-s − 729·27-s + 7.63e3·29-s + 2.73e3·30-s + 966.·31-s + 8.19e3·32-s + ⋯
L(s)  = 1  − 1.49·2-s − 0.577·3-s + 1.22·4-s + 0.643·5-s + 0.862·6-s − 0.342·8-s + 0.333·9-s − 0.961·10-s + 0.736·11-s − 0.709·12-s + 1.88·13-s − 0.371·15-s − 0.718·16-s − 0.866·17-s − 0.497·18-s − 1.33·19-s + 0.791·20-s − 1.09·22-s − 0.252·23-s + 0.197·24-s − 0.585·25-s − 2.81·26-s − 0.192·27-s + 1.68·29-s + 0.555·30-s + 0.180·31-s + 1.41·32-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: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8513347875\)
\(L(\frac12)\) \(\approx\) \(0.8513347875\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.44T + 32T^{2} \)
5 \( 1 - 36T + 3.12e3T^{2} \)
11 \( 1 - 295.T + 1.61e5T^{2} \)
13 \( 1 - 1.14e3T + 3.71e5T^{2} \)
17 \( 1 + 1.03e3T + 1.41e6T^{2} \)
19 \( 1 + 2.10e3T + 2.47e6T^{2} \)
23 \( 1 + 640.T + 6.43e6T^{2} \)
29 \( 1 - 7.63e3T + 2.05e7T^{2} \)
31 \( 1 - 966.T + 2.86e7T^{2} \)
37 \( 1 + 1.77e3T + 6.93e7T^{2} \)
41 \( 1 - 1.19e4T + 1.15e8T^{2} \)
43 \( 1 + 1.98e4T + 1.47e8T^{2} \)
47 \( 1 - 2.79e4T + 2.29e8T^{2} \)
53 \( 1 + 7.11e3T + 4.18e8T^{2} \)
59 \( 1 - 2.08e4T + 7.14e8T^{2} \)
61 \( 1 - 2.38e4T + 8.44e8T^{2} \)
67 \( 1 - 3.46e4T + 1.35e9T^{2} \)
71 \( 1 + 2.84e4T + 1.80e9T^{2} \)
73 \( 1 - 1.52e4T + 2.07e9T^{2} \)
79 \( 1 + 7.30e4T + 3.07e9T^{2} \)
83 \( 1 - 3.03e4T + 3.93e9T^{2} \)
89 \( 1 - 3.60e4T + 5.58e9T^{2} \)
97 \( 1 - 1.53e5T + 8.58e9T^{2} \)
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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

−11.67161402015328850697049868118, −10.83297079356045490295806825367, −10.11836501561123996283428693267, −8.953705187776048323470145902575, −8.350334454298132775655542657808, −6.72840305859930952293190095179, −6.10798913122270606452282899625, −4.21902896428097380529159442164, −1.95607142238872516671517008718, −0.814533279844596986429365072054, 0.814533279844596986429365072054, 1.95607142238872516671517008718, 4.21902896428097380529159442164, 6.10798913122270606452282899625, 6.72840305859930952293190095179, 8.350334454298132775655542657808, 8.953705187776048323470145902575, 10.11836501561123996283428693267, 10.83297079356045490295806825367, 11.67161402015328850697049868118

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