Properties

Label 2-2009-1.1-c1-0-117
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  + 2.36·2-s + 1.64·3-s + 3.61·4-s + 4.02·5-s + 3.90·6-s + 3.82·8-s − 0.289·9-s + 9.54·10-s − 2.19·11-s + 5.95·12-s − 2.49·13-s + 6.63·15-s + 1.83·16-s − 4.73·17-s − 0.686·18-s − 0.560·19-s + 14.5·20-s − 5.21·22-s + 7.86·23-s + 6.29·24-s + 11.2·25-s − 5.90·26-s − 5.41·27-s + 1.33·29-s + 15.7·30-s − 4.18·31-s − 3.30·32-s + ⋯
L(s)  = 1  + 1.67·2-s + 0.950·3-s + 1.80·4-s + 1.80·5-s + 1.59·6-s + 1.35·8-s − 0.0965·9-s + 3.01·10-s − 0.663·11-s + 1.71·12-s − 0.691·13-s + 1.71·15-s + 0.458·16-s − 1.14·17-s − 0.161·18-s − 0.128·19-s + 3.25·20-s − 1.11·22-s + 1.64·23-s + 1.28·24-s + 2.24·25-s − 1.15·26-s − 1.04·27-s + 0.247·29-s + 2.87·30-s − 0.752·31-s − 0.583·32-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

\(L(1)\) \(\approx\) \(7.638318588\)
\(L(\frac12)\) \(\approx\) \(7.638318588\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.36T + 2T^{2} \)
3 \( 1 - 1.64T + 3T^{2} \)
5 \( 1 - 4.02T + 5T^{2} \)
11 \( 1 + 2.19T + 11T^{2} \)
13 \( 1 + 2.49T + 13T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 + 0.560T + 19T^{2} \)
23 \( 1 - 7.86T + 23T^{2} \)
29 \( 1 - 1.33T + 29T^{2} \)
31 \( 1 + 4.18T + 31T^{2} \)
37 \( 1 - 8.64T + 37T^{2} \)
43 \( 1 - 2.23T + 43T^{2} \)
47 \( 1 + 3.19T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 6.40T + 61T^{2} \)
67 \( 1 - 0.699T + 67T^{2} \)
71 \( 1 + 5.49T + 71T^{2} \)
73 \( 1 + 6.06T + 73T^{2} \)
79 \( 1 - 0.250T + 79T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + 5.57T + 89T^{2} \)
97 \( 1 + 9.96T + 97T^{2} \)
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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

−9.238865391602667796023086244255, −8.456503430906380966996776014928, −7.22437225619979663407300271074, −6.57140812061434849434473115792, −5.71413404159530468416935350636, −5.16387934285276663112957341758, −4.37029974080644057579523836534, −3.00414447719417055820912909186, −2.61645797028441105454757771658, −1.87626123527978726003228173848, 1.87626123527978726003228173848, 2.61645797028441105454757771658, 3.00414447719417055820912909186, 4.37029974080644057579523836534, 5.16387934285276663112957341758, 5.71413404159530468416935350636, 6.57140812061434849434473115792, 7.22437225619979663407300271074, 8.456503430906380966996776014928, 9.238865391602667796023086244255

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