Properties

Label 2-2151-1.1-c3-0-138
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $126.913$
Root an. cond. $11.2655$
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  + 0.483·2-s − 7.76·4-s + 20.7·5-s + 33.2·7-s − 7.62·8-s + 10.0·10-s − 3.33·11-s − 19.7·13-s + 16.0·14-s + 58.4·16-s − 118.·17-s − 16.5·19-s − 161.·20-s − 1.61·22-s + 15.2·23-s + 306.·25-s − 9.53·26-s − 258.·28-s − 95.1·29-s + 279.·31-s + 89.2·32-s − 57.3·34-s + 691.·35-s − 10.2·37-s − 7.99·38-s − 158.·40-s − 353.·41-s + ⋯
L(s)  = 1  + 0.170·2-s − 0.970·4-s + 1.85·5-s + 1.79·7-s − 0.336·8-s + 0.317·10-s − 0.0914·11-s − 0.420·13-s + 0.306·14-s + 0.913·16-s − 1.69·17-s − 0.199·19-s − 1.80·20-s − 0.0156·22-s + 0.138·23-s + 2.45·25-s − 0.0719·26-s − 1.74·28-s − 0.609·29-s + 1.61·31-s + 0.492·32-s − 0.289·34-s + 3.33·35-s − 0.0456·37-s − 0.0341·38-s − 0.625·40-s − 1.34·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(126.913\)
Root analytic conductor: \(11.2655\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.508009564\)
\(L(\frac12)\) \(\approx\) \(3.508009564\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 - 239T \)
good2 \( 1 - 0.483T + 8T^{2} \)
5 \( 1 - 20.7T + 125T^{2} \)
7 \( 1 - 33.2T + 343T^{2} \)
11 \( 1 + 3.33T + 1.33e3T^{2} \)
13 \( 1 + 19.7T + 2.19e3T^{2} \)
17 \( 1 + 118.T + 4.91e3T^{2} \)
19 \( 1 + 16.5T + 6.85e3T^{2} \)
23 \( 1 - 15.2T + 1.21e4T^{2} \)
29 \( 1 + 95.1T + 2.43e4T^{2} \)
31 \( 1 - 279.T + 2.97e4T^{2} \)
37 \( 1 + 10.2T + 5.06e4T^{2} \)
41 \( 1 + 353.T + 6.89e4T^{2} \)
43 \( 1 - 116.T + 7.95e4T^{2} \)
47 \( 1 - 295.T + 1.03e5T^{2} \)
53 \( 1 - 631.T + 1.48e5T^{2} \)
59 \( 1 - 444.T + 2.05e5T^{2} \)
61 \( 1 + 612.T + 2.26e5T^{2} \)
67 \( 1 - 835.T + 3.00e5T^{2} \)
71 \( 1 - 509.T + 3.57e5T^{2} \)
73 \( 1 + 23.9T + 3.89e5T^{2} \)
79 \( 1 - 597.T + 4.93e5T^{2} \)
83 \( 1 - 653.T + 5.71e5T^{2} \)
89 \( 1 + 1.40e3T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−8.717205101853525017239152532965, −8.292636968760520371381710552234, −7.09408023466242780940915142669, −6.17722169220629000489937402096, −5.29750524188987296698006474736, −4.91310720939631195188198027658, −4.16151921567522902922660550014, −2.51272270605824457804474329960, −1.90497689318012120592709670447, −0.868088791452760325300531136428, 0.868088791452760325300531136428, 1.90497689318012120592709670447, 2.51272270605824457804474329960, 4.16151921567522902922660550014, 4.91310720939631195188198027658, 5.29750524188987296698006474736, 6.17722169220629000489937402096, 7.09408023466242780940915142669, 8.292636968760520371381710552234, 8.717205101853525017239152532965

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