Properties

Label 2-147-1.1-c5-0-12
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  − 9.38·2-s − 9·3-s + 56.1·4-s − 71.7·5-s + 84.5·6-s − 226.·8-s + 81·9-s + 673.·10-s − 560.·11-s − 505.·12-s + 533.·13-s + 645.·15-s + 333.·16-s + 1.00e3·17-s − 760.·18-s + 1.36e3·19-s − 4.02e3·20-s + 5.26e3·22-s + 3.22e3·23-s + 2.04e3·24-s + 2.02e3·25-s − 5.00e3·26-s − 729·27-s − 753.·29-s − 6.06e3·30-s + 8.20e3·31-s + 4.12e3·32-s + ⋯
L(s)  = 1  − 1.65·2-s − 0.577·3-s + 1.75·4-s − 1.28·5-s + 0.958·6-s − 1.25·8-s + 0.333·9-s + 2.12·10-s − 1.39·11-s − 1.01·12-s + 0.875·13-s + 0.740·15-s + 0.325·16-s + 0.844·17-s − 0.553·18-s + 0.869·19-s − 2.25·20-s + 2.31·22-s + 1.27·23-s + 0.723·24-s + 0.646·25-s − 1.45·26-s − 0.192·27-s − 0.166·29-s − 1.22·30-s + 1.53·31-s + 0.712·32-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 \( 1 + 9.38T + 32T^{2} \)
5 \( 1 + 71.7T + 3.12e3T^{2} \)
11 \( 1 + 560.T + 1.61e5T^{2} \)
13 \( 1 - 533.T + 3.71e5T^{2} \)
17 \( 1 - 1.00e3T + 1.41e6T^{2} \)
19 \( 1 - 1.36e3T + 2.47e6T^{2} \)
23 \( 1 - 3.22e3T + 6.43e6T^{2} \)
29 \( 1 + 753.T + 2.05e7T^{2} \)
31 \( 1 - 8.20e3T + 2.86e7T^{2} \)
37 \( 1 + 2.80e3T + 6.93e7T^{2} \)
41 \( 1 - 245.T + 1.15e8T^{2} \)
43 \( 1 + 1.75e4T + 1.47e8T^{2} \)
47 \( 1 + 1.63e4T + 2.29e8T^{2} \)
53 \( 1 + 2.96e4T + 4.18e8T^{2} \)
59 \( 1 + 1.03e4T + 7.14e8T^{2} \)
61 \( 1 - 954.T + 8.44e8T^{2} \)
67 \( 1 + 1.98e4T + 1.35e9T^{2} \)
71 \( 1 - 6.21e4T + 1.80e9T^{2} \)
73 \( 1 - 2.71e4T + 2.07e9T^{2} \)
79 \( 1 - 4.46e4T + 3.07e9T^{2} \)
83 \( 1 - 1.56e4T + 3.93e9T^{2} \)
89 \( 1 - 1.36e4T + 5.58e9T^{2} \)
97 \( 1 + 1.29e4T + 8.58e9T^{2} \)
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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

−11.28552048779940648937517277410, −10.62756941014156541219263681440, −9.622740497705371199578976456630, −8.224738214841194984616247590072, −7.80523390741995263881636308002, −6.69067382447224610595070230210, −5.03780920621466594659905877487, −3.16910514019611927254661152818, −1.10574336901116235880305312221, 0, 1.10574336901116235880305312221, 3.16910514019611927254661152818, 5.03780920621466594659905877487, 6.69067382447224610595070230210, 7.80523390741995263881636308002, 8.224738214841194984616247590072, 9.622740497705371199578976456630, 10.62756941014156541219263681440, 11.28552048779940648937517277410

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