Properties

Label 2-147-1.1-c5-0-28
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  + 10.0·2-s + 9·3-s + 68.1·4-s + 70.3·5-s + 90.0·6-s + 362.·8-s + 81·9-s + 704.·10-s − 731.·11-s + 613.·12-s + 899.·13-s + 633.·15-s + 1.44e3·16-s − 1.39e3·17-s + 810.·18-s − 190.·19-s + 4.79e3·20-s − 7.32e3·22-s + 42.9·23-s + 3.25e3·24-s + 1.82e3·25-s + 9.00e3·26-s + 729·27-s − 7.74e3·29-s + 6.33e3·30-s + 1.17e3·31-s + 2.85e3·32-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.577·3-s + 2.13·4-s + 1.25·5-s + 1.02·6-s + 2.00·8-s + 0.333·9-s + 2.22·10-s − 1.82·11-s + 1.23·12-s + 1.47·13-s + 0.726·15-s + 1.40·16-s − 1.16·17-s + 0.589·18-s − 0.120·19-s + 2.68·20-s − 3.22·22-s + 0.0169·23-s + 1.15·24-s + 0.583·25-s + 2.61·26-s + 0.192·27-s − 1.71·29-s + 1.28·30-s + 0.220·31-s + 0.492·32-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: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(7.593944550\)
\(L(\frac12)\) \(\approx\) \(7.593944550\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.0T + 32T^{2} \)
5 \( 1 - 70.3T + 3.12e3T^{2} \)
11 \( 1 + 731.T + 1.61e5T^{2} \)
13 \( 1 - 899.T + 3.71e5T^{2} \)
17 \( 1 + 1.39e3T + 1.41e6T^{2} \)
19 \( 1 + 190.T + 2.47e6T^{2} \)
23 \( 1 - 42.9T + 6.43e6T^{2} \)
29 \( 1 + 7.74e3T + 2.05e7T^{2} \)
31 \( 1 - 1.17e3T + 2.86e7T^{2} \)
37 \( 1 - 9.28e3T + 6.93e7T^{2} \)
41 \( 1 - 1.34e4T + 1.15e8T^{2} \)
43 \( 1 - 6.03e3T + 1.47e8T^{2} \)
47 \( 1 + 3.24e3T + 2.29e8T^{2} \)
53 \( 1 - 2.56e4T + 4.18e8T^{2} \)
59 \( 1 + 2.64e4T + 7.14e8T^{2} \)
61 \( 1 + 6.43e3T + 8.44e8T^{2} \)
67 \( 1 + 2.38e4T + 1.35e9T^{2} \)
71 \( 1 + 4.46e4T + 1.80e9T^{2} \)
73 \( 1 + 3.96e4T + 2.07e9T^{2} \)
79 \( 1 - 2.31e4T + 3.07e9T^{2} \)
83 \( 1 + 1.72e4T + 3.93e9T^{2} \)
89 \( 1 - 4.99e4T + 5.58e9T^{2} \)
97 \( 1 + 1.68e4T + 8.58e9T^{2} \)
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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

−12.89788777813784850596785384575, −11.20325758307028494722933586657, −10.48696662289040963713905720170, −9.058336243079591994006273234150, −7.59795075351536710752835566600, −6.19664887741897577890375479623, −5.53535069728199807220933163874, −4.25407851635170220253298790363, −2.84420823570864719464230548146, −1.97136005584661824083699038581, 1.97136005584661824083699038581, 2.84420823570864719464230548146, 4.25407851635170220253298790363, 5.53535069728199807220933163874, 6.19664887741897577890375479623, 7.59795075351536710752835566600, 9.058336243079591994006273234150, 10.48696662289040963713905720170, 11.20325758307028494722933586657, 12.89788777813784850596785384575

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