Properties

Label 2-2151-1.1-c3-0-183
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.58·2-s + 13.0·4-s + 15.1·5-s − 25.8·7-s − 23.2·8-s − 69.4·10-s − 0.294·11-s + 3.35·13-s + 118.·14-s + 2.01·16-s − 39.1·17-s + 11.2·19-s + 197.·20-s + 1.35·22-s − 44.6·23-s + 103.·25-s − 15.3·26-s − 337.·28-s − 75.9·29-s − 34.1·31-s + 176.·32-s + 179.·34-s − 390.·35-s + 114.·37-s − 51.3·38-s − 350.·40-s + 364.·41-s + ⋯
L(s)  = 1  − 1.62·2-s + 1.63·4-s + 1.35·5-s − 1.39·7-s − 1.02·8-s − 2.19·10-s − 0.00808·11-s + 0.0715·13-s + 2.26·14-s + 0.0314·16-s − 0.558·17-s + 0.135·19-s + 2.20·20-s + 0.0131·22-s − 0.404·23-s + 0.830·25-s − 0.116·26-s − 2.27·28-s − 0.486·29-s − 0.197·31-s + 0.974·32-s + 0.906·34-s − 1.88·35-s + 0.509·37-s − 0.219·38-s − 1.38·40-s + 1.38·41-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: \(1\)
Selberg data: \((2,\ 2151,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.58T + 8T^{2} \)
5 \( 1 - 15.1T + 125T^{2} \)
7 \( 1 + 25.8T + 343T^{2} \)
11 \( 1 + 0.294T + 1.33e3T^{2} \)
13 \( 1 - 3.35T + 2.19e3T^{2} \)
17 \( 1 + 39.1T + 4.91e3T^{2} \)
19 \( 1 - 11.2T + 6.85e3T^{2} \)
23 \( 1 + 44.6T + 1.21e4T^{2} \)
29 \( 1 + 75.9T + 2.43e4T^{2} \)
31 \( 1 + 34.1T + 2.97e4T^{2} \)
37 \( 1 - 114.T + 5.06e4T^{2} \)
41 \( 1 - 364.T + 6.89e4T^{2} \)
43 \( 1 - 145.T + 7.95e4T^{2} \)
47 \( 1 - 322.T + 1.03e5T^{2} \)
53 \( 1 - 345.T + 1.48e5T^{2} \)
59 \( 1 - 701.T + 2.05e5T^{2} \)
61 \( 1 + 357.T + 2.26e5T^{2} \)
67 \( 1 + 748.T + 3.00e5T^{2} \)
71 \( 1 + 730.T + 3.57e5T^{2} \)
73 \( 1 - 41.7T + 3.89e5T^{2} \)
79 \( 1 - 450.T + 4.93e5T^{2} \)
83 \( 1 - 826.T + 5.71e5T^{2} \)
89 \( 1 + 1.29e3T + 7.04e5T^{2} \)
97 \( 1 + 533.T + 9.12e5T^{2} \)
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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

−8.684114184353100606182954682704, −7.60807809515235387108996750985, −6.89952732464799639446294289103, −6.18280990643193031924318556282, −5.64094219799653069204810173569, −4.13421468852126607864079218227, −2.78372749459842987293565115744, −2.14936356974835570214024282739, −1.05657662754521387367535002743, 0, 1.05657662754521387367535002743, 2.14936356974835570214024282739, 2.78372749459842987293565115744, 4.13421468852126607864079218227, 5.64094219799653069204810173569, 6.18280990643193031924318556282, 6.89952732464799639446294289103, 7.60807809515235387108996750985, 8.684114184353100606182954682704

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