Properties

Label 2-2009-1.1-c1-0-20
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  − 0.656·2-s − 1.65·3-s − 1.56·4-s + 1.08·6-s + 2.34·8-s − 0.255·9-s + 1.82·11-s + 2.59·12-s + 2.59·13-s + 1.59·16-s − 0.568·17-s + 0.167·18-s + 2.74·19-s − 1.19·22-s − 5.48·23-s − 3.88·24-s − 5·25-s − 1.70·26-s + 5.39·27-s − 1.31·29-s − 9.64·31-s − 5.73·32-s − 3.02·33-s + 0.373·34-s + 0.401·36-s − 3.40·37-s − 1.80·38-s + ⋯
L(s)  = 1  − 0.464·2-s − 0.956·3-s − 0.784·4-s + 0.444·6-s + 0.828·8-s − 0.0852·9-s + 0.550·11-s + 0.750·12-s + 0.720·13-s + 0.399·16-s − 0.137·17-s + 0.0395·18-s + 0.629·19-s − 0.255·22-s − 1.14·23-s − 0.792·24-s − 25-s − 0.334·26-s + 1.03·27-s − 0.243·29-s − 1.73·31-s − 1.01·32-s − 0.526·33-s + 0.0640·34-s + 0.0668·36-s − 0.559·37-s − 0.292·38-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

\(L(1)\) \(\approx\) \(0.5946172332\)
\(L(\frac12)\) \(\approx\) \(0.5946172332\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.656T + 2T^{2} \)
3 \( 1 + 1.65T + 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 1.82T + 11T^{2} \)
13 \( 1 - 2.59T + 13T^{2} \)
17 \( 1 + 0.568T + 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + 5.48T + 23T^{2} \)
29 \( 1 + 1.31T + 29T^{2} \)
31 \( 1 + 9.64T + 31T^{2} \)
37 \( 1 + 3.40T + 37T^{2} \)
43 \( 1 - 2.56T + 43T^{2} \)
47 \( 1 - 3.73T + 47T^{2} \)
53 \( 1 - 1.82T + 53T^{2} \)
59 \( 1 - 6.33T + 59T^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 6.51T + 71T^{2} \)
73 \( 1 + 5.19T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 1.13T + 83T^{2} \)
89 \( 1 - 2.59T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−9.128093667079158757980263425504, −8.514597161438106273538148084324, −7.67536256495095490403583237224, −6.78902796673198617108705211862, −5.74101432670967565354028464396, −5.41582365237570744644510117108, −4.22248495341417712037576700795, −3.58195727433523434818337767375, −1.82744876008328932777988601048, −0.58950026196098288091025239744, 0.58950026196098288091025239744, 1.82744876008328932777988601048, 3.58195727433523434818337767375, 4.22248495341417712037576700795, 5.41582365237570744644510117108, 5.74101432670967565354028464396, 6.78902796673198617108705211862, 7.67536256495095490403583237224, 8.514597161438106273538148084324, 9.128093667079158757980263425504

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