Properties

Label 2-2009-1.1-c1-0-119
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  + 2.03·2-s − 3.03·3-s + 2.12·4-s + 3.82·5-s − 6.15·6-s + 0.255·8-s + 6.18·9-s + 7.77·10-s − 5.96·11-s − 6.44·12-s − 1.44·13-s − 11.6·15-s − 3.73·16-s − 6.06·17-s + 12.5·18-s + 0.0743·19-s + 8.13·20-s − 12.1·22-s − 4.43·23-s − 0.774·24-s + 9.64·25-s − 2.93·26-s − 9.66·27-s − 1.92·29-s − 23.5·30-s − 1.76·31-s − 8.09·32-s + ⋯
L(s)  = 1  + 1.43·2-s − 1.75·3-s + 1.06·4-s + 1.71·5-s − 2.51·6-s + 0.0903·8-s + 2.06·9-s + 2.45·10-s − 1.79·11-s − 1.86·12-s − 0.400·13-s − 2.99·15-s − 0.933·16-s − 1.47·17-s + 2.96·18-s + 0.0170·19-s + 1.81·20-s − 2.58·22-s − 0.924·23-s − 0.158·24-s + 1.92·25-s − 0.575·26-s − 1.85·27-s − 0.357·29-s − 4.30·30-s − 0.316·31-s − 1.43·32-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 \( 1 - 2.03T + 2T^{2} \)
3 \( 1 + 3.03T + 3T^{2} \)
5 \( 1 - 3.82T + 5T^{2} \)
11 \( 1 + 5.96T + 11T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 + 6.06T + 17T^{2} \)
19 \( 1 - 0.0743T + 19T^{2} \)
23 \( 1 + 4.43T + 23T^{2} \)
29 \( 1 + 1.92T + 29T^{2} \)
31 \( 1 + 1.76T + 31T^{2} \)
37 \( 1 - 0.497T + 37T^{2} \)
43 \( 1 - 4.10T + 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 2.94T + 61T^{2} \)
67 \( 1 + 1.12T + 67T^{2} \)
71 \( 1 - 5.87T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 0.670T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−9.007668117285132455784646579807, −7.48040931734112700231572239950, −6.60680714468579339511119276325, −5.96765681364843940642052207411, −5.57186786180215759854275337691, −4.95342665838951298116840952704, −4.34545966809654338741183507500, −2.69480238715705784688559381245, −1.94216982177035905095460641142, 0, 1.94216982177035905095460641142, 2.69480238715705784688559381245, 4.34545966809654338741183507500, 4.95342665838951298116840952704, 5.57186786180215759854275337691, 5.96765681364843940642052207411, 6.60680714468579339511119276325, 7.48040931734112700231572239950, 9.007668117285132455784646579807

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