Properties

Label 2-147-1.1-c3-0-6
Degree $2$
Conductor $147$
Sign $1$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.248·2-s + 3·3-s − 7.93·4-s + 12.4·5-s + 0.744·6-s − 3.95·8-s + 9·9-s + 3.08·10-s + 60.3·11-s − 23.8·12-s − 36.4·13-s + 37.3·15-s + 62.5·16-s + 48.7·17-s + 2.23·18-s + 50.5·19-s − 98.7·20-s + 14.9·22-s + 138.·23-s − 11.8·24-s + 29.6·25-s − 9.03·26-s + 27·27-s − 61.1·29-s + 9.25·30-s + 1.16·31-s + 47.1·32-s + ⋯
L(s)  = 1  + 0.0877·2-s + 0.577·3-s − 0.992·4-s + 1.11·5-s + 0.0506·6-s − 0.174·8-s + 0.333·9-s + 0.0975·10-s + 1.65·11-s − 0.572·12-s − 0.777·13-s + 0.642·15-s + 0.976·16-s + 0.695·17-s + 0.0292·18-s + 0.610·19-s − 1.10·20-s + 0.144·22-s + 1.25·23-s − 0.100·24-s + 0.236·25-s − 0.0681·26-s + 0.192·27-s − 0.391·29-s + 0.0563·30-s + 0.00677·31-s + 0.260·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.169924848\)
\(L(\frac12)\) \(\approx\) \(2.169924848\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
7 \( 1 \)
good2 \( 1 - 0.248T + 8T^{2} \)
5 \( 1 - 12.4T + 125T^{2} \)
11 \( 1 - 60.3T + 1.33e3T^{2} \)
13 \( 1 + 36.4T + 2.19e3T^{2} \)
17 \( 1 - 48.7T + 4.91e3T^{2} \)
19 \( 1 - 50.5T + 6.85e3T^{2} \)
23 \( 1 - 138.T + 1.21e4T^{2} \)
29 \( 1 + 61.1T + 2.43e4T^{2} \)
31 \( 1 - 1.16T + 2.97e4T^{2} \)
37 \( 1 - 69.5T + 5.06e4T^{2} \)
41 \( 1 + 308.T + 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 + 389.T + 1.03e5T^{2} \)
53 \( 1 - 314.T + 1.48e5T^{2} \)
59 \( 1 + 844.T + 2.05e5T^{2} \)
61 \( 1 - 338.T + 2.26e5T^{2} \)
67 \( 1 + 971.T + 3.00e5T^{2} \)
71 \( 1 + 98.4T + 3.57e5T^{2} \)
73 \( 1 + 710.T + 3.89e5T^{2} \)
79 \( 1 + 486.T + 4.93e5T^{2} \)
83 \( 1 + 605.T + 5.71e5T^{2} \)
89 \( 1 + 218.T + 7.04e5T^{2} \)
97 \( 1 - 782.T + 9.12e5T^{2} \)
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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

−12.85015961083837615760210855516, −11.78704272531341809092773304235, −10.08028635030759481651812866391, −9.442915390485614656877258437934, −8.779043722867280340775853018624, −7.29139900056553300319862237374, −5.90997829966740273728624708401, −4.68891351236906856885271221368, −3.28047563047666095708959809526, −1.39459218354390022131889674103, 1.39459218354390022131889674103, 3.28047563047666095708959809526, 4.68891351236906856885271221368, 5.90997829966740273728624708401, 7.29139900056553300319862237374, 8.779043722867280340775853018624, 9.442915390485614656877258437934, 10.08028635030759481651812866391, 11.78704272531341809092773304235, 12.85015961083837615760210855516

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