Properties

Label 2-147-1.1-c5-0-16
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  − 3.09·2-s − 9·3-s − 22.4·4-s − 13.7·5-s + 27.8·6-s + 168.·8-s + 81·9-s + 42.6·10-s − 3.66·11-s + 201.·12-s + 780.·13-s + 124.·15-s + 197.·16-s − 50.0·17-s − 250.·18-s − 1.06e3·19-s + 309.·20-s + 11.3·22-s + 4.10e3·23-s − 1.51e3·24-s − 2.93e3·25-s − 2.41e3·26-s − 729·27-s − 1.48e3·29-s − 383.·30-s − 5.51e3·31-s − 5.99e3·32-s + ⋯
L(s)  = 1  − 0.546·2-s − 0.577·3-s − 0.701·4-s − 0.246·5-s + 0.315·6-s + 0.929·8-s + 0.333·9-s + 0.134·10-s − 0.00913·11-s + 0.404·12-s + 1.28·13-s + 0.142·15-s + 0.193·16-s − 0.0420·17-s − 0.182·18-s − 0.675·19-s + 0.173·20-s + 0.00499·22-s + 1.61·23-s − 0.536·24-s − 0.939·25-s − 0.699·26-s − 0.192·27-s − 0.328·29-s − 0.0778·30-s − 1.03·31-s − 1.03·32-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: Trivial
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 \( 1 + 3.09T + 32T^{2} \)
5 \( 1 + 13.7T + 3.12e3T^{2} \)
11 \( 1 + 3.66T + 1.61e5T^{2} \)
13 \( 1 - 780.T + 3.71e5T^{2} \)
17 \( 1 + 50.0T + 1.41e6T^{2} \)
19 \( 1 + 1.06e3T + 2.47e6T^{2} \)
23 \( 1 - 4.10e3T + 6.43e6T^{2} \)
29 \( 1 + 1.48e3T + 2.05e7T^{2} \)
31 \( 1 + 5.51e3T + 2.86e7T^{2} \)
37 \( 1 - 6.14e3T + 6.93e7T^{2} \)
41 \( 1 + 1.07e4T + 1.15e8T^{2} \)
43 \( 1 - 1.76e4T + 1.47e8T^{2} \)
47 \( 1 + 2.94e4T + 2.29e8T^{2} \)
53 \( 1 + 1.92e4T + 4.18e8T^{2} \)
59 \( 1 + 6.61e3T + 7.14e8T^{2} \)
61 \( 1 + 3.67e4T + 8.44e8T^{2} \)
67 \( 1 - 4.69e4T + 1.35e9T^{2} \)
71 \( 1 + 4.16e4T + 1.80e9T^{2} \)
73 \( 1 + 2.94e4T + 2.07e9T^{2} \)
79 \( 1 - 2.21e4T + 3.07e9T^{2} \)
83 \( 1 - 3.89e3T + 3.93e9T^{2} \)
89 \( 1 + 2.05e4T + 5.58e9T^{2} \)
97 \( 1 + 1.77e4T + 8.58e9T^{2} \)
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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

−11.33419784147024675686079240586, −10.69920217499525129086630180848, −9.485592701395620817536924369972, −8.603997388367433794239898482231, −7.52702408714046405595038270924, −6.16700345516481352593222740902, −4.88542735046437283360236734404, −3.69707694097426592562361961614, −1.36273716593712638291495821652, 0, 1.36273716593712638291495821652, 3.69707694097426592562361961614, 4.88542735046437283360236734404, 6.16700345516481352593222740902, 7.52702408714046405595038270924, 8.603997388367433794239898482231, 9.485592701395620817536924369972, 10.69920217499525129086630180848, 11.33419784147024675686079240586

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