Properties

Label 2-2151-1.1-c1-0-53
Degree $2$
Conductor $2151$
Sign $-1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.67·2-s + 5.15·4-s − 0.519·5-s + 1.02·7-s − 8.42·8-s + 1.38·10-s − 5.14·11-s − 2.29·13-s − 2.73·14-s + 12.2·16-s + 1.92·17-s + 3.15·19-s − 2.67·20-s + 13.7·22-s + 6.68·23-s − 4.73·25-s + 6.13·26-s + 5.25·28-s + 7.68·29-s − 1.20·31-s − 15.8·32-s − 5.13·34-s − 0.530·35-s − 0.791·37-s − 8.42·38-s + 4.37·40-s − 2.49·41-s + ⋯
L(s)  = 1  − 1.89·2-s + 2.57·4-s − 0.232·5-s + 0.385·7-s − 2.97·8-s + 0.439·10-s − 1.55·11-s − 0.636·13-s − 0.729·14-s + 3.05·16-s + 0.466·17-s + 0.722·19-s − 0.597·20-s + 2.93·22-s + 1.39·23-s − 0.946·25-s + 1.20·26-s + 0.993·28-s + 1.42·29-s − 0.216·31-s − 2.80·32-s − 0.881·34-s − 0.0895·35-s − 0.130·37-s − 1.36·38-s + 0.691·40-s − 0.389·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + 2.67T + 2T^{2} \)
5 \( 1 + 0.519T + 5T^{2} \)
7 \( 1 - 1.02T + 7T^{2} \)
11 \( 1 + 5.14T + 11T^{2} \)
13 \( 1 + 2.29T + 13T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + 1.20T + 31T^{2} \)
37 \( 1 + 0.791T + 37T^{2} \)
41 \( 1 + 2.49T + 41T^{2} \)
43 \( 1 - 1.15T + 43T^{2} \)
47 \( 1 + 8.92T + 47T^{2} \)
53 \( 1 - 7.62T + 53T^{2} \)
59 \( 1 + 9.24T + 59T^{2} \)
61 \( 1 - 2.23T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 - 4.46T + 71T^{2} \)
73 \( 1 + 2.29T + 73T^{2} \)
79 \( 1 + 9.20T + 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + 7.81T + 89T^{2} \)
97 \( 1 + 4.25T + 97T^{2} \)
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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

−8.446482033757533056940452304938, −8.159537188143234298063418752883, −7.38716275487410493904879601016, −6.87582965283516068352160110217, −5.67200805944442821460722738715, −4.89448782145485875045772732730, −3.15488960420479644611664137727, −2.46248048932991588836976287666, −1.26018766676849709596599869910, 0, 1.26018766676849709596599869910, 2.46248048932991588836976287666, 3.15488960420479644611664137727, 4.89448782145485875045772732730, 5.67200805944442821460722738715, 6.87582965283516068352160110217, 7.38716275487410493904879601016, 8.159537188143234298063418752883, 8.446482033757533056940452304938

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