Properties

Label 2-2009-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.822·2-s − 0.955·3-s − 1.32·4-s − 3.05·5-s − 0.785·6-s − 2.73·8-s − 2.08·9-s − 2.51·10-s − 6.35·11-s + 1.26·12-s + 3.93·13-s + 2.91·15-s + 0.402·16-s − 2.97·17-s − 1.71·18-s − 5.53·19-s + 4.04·20-s − 5.22·22-s − 5.46·23-s + 2.60·24-s + 4.34·25-s + 3.23·26-s + 4.85·27-s − 5.86·29-s + 2.39·30-s + 10.6·31-s + 5.79·32-s + ⋯
L(s)  = 1  + 0.581·2-s − 0.551·3-s − 0.662·4-s − 1.36·5-s − 0.320·6-s − 0.966·8-s − 0.695·9-s − 0.794·10-s − 1.91·11-s + 0.365·12-s + 1.09·13-s + 0.753·15-s + 0.100·16-s − 0.720·17-s − 0.404·18-s − 1.27·19-s + 0.905·20-s − 1.11·22-s − 1.13·23-s + 0.532·24-s + 0.868·25-s + 0.633·26-s + 0.935·27-s − 1.08·29-s + 0.438·30-s + 1.91·31-s + 1.02·32-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2009,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2491397283\)
\(L(\frac12)\) \(\approx\) \(0.2491397283\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.822T + 2T^{2} \)
3 \( 1 + 0.955T + 3T^{2} \)
5 \( 1 + 3.05T + 5T^{2} \)
11 \( 1 + 6.35T + 11T^{2} \)
13 \( 1 - 3.93T + 13T^{2} \)
17 \( 1 + 2.97T + 17T^{2} \)
19 \( 1 + 5.53T + 19T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 + 5.86T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 + 0.638T + 37T^{2} \)
43 \( 1 + 0.978T + 43T^{2} \)
47 \( 1 + 3.60T + 47T^{2} \)
53 \( 1 + 2.10T + 53T^{2} \)
59 \( 1 + 5.71T + 59T^{2} \)
61 \( 1 - 6.61T + 61T^{2} \)
67 \( 1 - 2.82T + 67T^{2} \)
71 \( 1 + 0.220T + 71T^{2} \)
73 \( 1 + 6.59T + 73T^{2} \)
79 \( 1 - 1.73T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 2.25T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−8.858666677174237628494360091681, −8.196499017023542813599154139685, −7.943013529500692506857709213264, −6.52226815206138186153360471249, −5.86164472000291288233353722503, −4.99944730870728909330690883510, −4.32460599547909337663200137126, −3.53840792253679266612180168874, −2.55673224513536652692280122245, −0.29301267786317257652006075403, 0.29301267786317257652006075403, 2.55673224513536652692280122245, 3.53840792253679266612180168874, 4.32460599547909337663200137126, 4.99944730870728909330690883510, 5.86164472000291288233353722503, 6.52226815206138186153360471249, 7.943013529500692506857709213264, 8.196499017023542813599154139685, 8.858666677174237628494360091681

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