Properties

Label 2-2009-1.1-c1-0-84
Degree $2$
Conductor $2009$
Sign $-1$
Analytic cond. $16.0419$
Root an. cond. $4.00523$
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  + 1.08·2-s − 2.08·3-s − 0.817·4-s − 0.209·5-s − 2.26·6-s − 3.06·8-s + 1.35·9-s − 0.227·10-s + 6.03·11-s + 1.70·12-s − 3.67·13-s + 0.437·15-s − 1.69·16-s + 5.37·17-s + 1.47·18-s + 3.54·19-s + 0.171·20-s + 6.56·22-s − 1.30·23-s + 6.39·24-s − 4.95·25-s − 3.99·26-s + 3.43·27-s − 8.00·29-s + 0.475·30-s + 0.384·31-s + 4.28·32-s + ⋯
L(s)  = 1  + 0.768·2-s − 1.20·3-s − 0.408·4-s − 0.0937·5-s − 0.926·6-s − 1.08·8-s + 0.452·9-s − 0.0720·10-s + 1.82·11-s + 0.492·12-s − 1.01·13-s + 0.112·15-s − 0.423·16-s + 1.30·17-s + 0.347·18-s + 0.812·19-s + 0.0383·20-s + 1.39·22-s − 0.271·23-s + 1.30·24-s − 0.991·25-s − 0.782·26-s + 0.660·27-s − 1.48·29-s + 0.0868·30-s + 0.0690·31-s + 0.757·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $-1$
Analytic conductor: \(16.0419\)
Root analytic conductor: \(4.00523\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2009,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 1.08T + 2T^{2} \)
3 \( 1 + 2.08T + 3T^{2} \)
5 \( 1 + 0.209T + 5T^{2} \)
11 \( 1 - 6.03T + 11T^{2} \)
13 \( 1 + 3.67T + 13T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 - 3.54T + 19T^{2} \)
23 \( 1 + 1.30T + 23T^{2} \)
29 \( 1 + 8.00T + 29T^{2} \)
31 \( 1 - 0.384T + 31T^{2} \)
37 \( 1 + 3.68T + 37T^{2} \)
43 \( 1 - 0.824T + 43T^{2} \)
47 \( 1 + 5.11T + 47T^{2} \)
53 \( 1 - 1.53T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 9.36T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + 7.77T + 73T^{2} \)
79 \( 1 + 6.04T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 0.520T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−9.001140029809659485794622359314, −7.78412429633665032311926291225, −6.95951543402134946435272789148, −5.99152865787834151278956160709, −5.62593883081304205267153379519, −4.77608002990278672198734818311, −3.97907488030448197110278937260, −3.17200756969222704890183264599, −1.40197895365906947914838220522, 0, 1.40197895365906947914838220522, 3.17200756969222704890183264599, 3.97907488030448197110278937260, 4.77608002990278672198734818311, 5.62593883081304205267153379519, 5.99152865787834151278956160709, 6.95951543402134946435272789148, 7.78412429633665032311926291225, 9.001140029809659485794622359314

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