Properties

Label 2-2009-1.1-c1-0-104
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  − 1.20·2-s + 0.200·3-s − 0.557·4-s + 3.21·5-s − 0.241·6-s + 3.07·8-s − 2.95·9-s − 3.86·10-s + 4.57·11-s − 0.112·12-s − 0.703·13-s + 0.646·15-s − 2.57·16-s − 4.25·17-s + 3.55·18-s − 8.04·19-s − 1.79·20-s − 5.49·22-s − 5.34·23-s + 0.617·24-s + 5.34·25-s + 0.844·26-s − 1.19·27-s − 5.39·29-s − 0.776·30-s − 7.61·31-s − 3.05·32-s + ⋯
L(s)  = 1  − 0.849·2-s + 0.116·3-s − 0.278·4-s + 1.43·5-s − 0.0985·6-s + 1.08·8-s − 0.986·9-s − 1.22·10-s + 1.38·11-s − 0.0323·12-s − 0.194·13-s + 0.166·15-s − 0.643·16-s − 1.03·17-s + 0.837·18-s − 1.84·19-s − 0.401·20-s − 1.17·22-s − 1.11·23-s + 0.126·24-s + 1.06·25-s + 0.165·26-s − 0.230·27-s − 1.00·29-s − 0.141·30-s − 1.36·31-s − 0.539·32-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 \( 1 + 1.20T + 2T^{2} \)
3 \( 1 - 0.200T + 3T^{2} \)
5 \( 1 - 3.21T + 5T^{2} \)
11 \( 1 - 4.57T + 11T^{2} \)
13 \( 1 + 0.703T + 13T^{2} \)
17 \( 1 + 4.25T + 17T^{2} \)
19 \( 1 + 8.04T + 19T^{2} \)
23 \( 1 + 5.34T + 23T^{2} \)
29 \( 1 + 5.39T + 29T^{2} \)
31 \( 1 + 7.61T + 31T^{2} \)
37 \( 1 - 5.19T + 37T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 - 7.73T + 59T^{2} \)
61 \( 1 - 2.48T + 61T^{2} \)
67 \( 1 + 3.09T + 67T^{2} \)
71 \( 1 - 5.11T + 71T^{2} \)
73 \( 1 + 4.13T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 4.90T + 83T^{2} \)
89 \( 1 + 5.97T + 89T^{2} \)
97 \( 1 + 1.45T + 97T^{2} \)
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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

−8.827573872594104747462913022870, −8.422819077790036112296869238294, −7.26769334926755478750715520763, −6.25868860726178528359821999494, −5.88501350241085958655476388276, −4.66540704997428635101415977902, −3.84654611908581700753496929413, −2.26178894877230404965869654861, −1.69192395454443625274249368119, 0, 1.69192395454443625274249368119, 2.26178894877230404965869654861, 3.84654611908581700753496929413, 4.66540704997428635101415977902, 5.88501350241085958655476388276, 6.25868860726178528359821999494, 7.26769334926755478750715520763, 8.422819077790036112296869238294, 8.827573872594104747462913022870

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