Properties

Label 2-2009-1.1-c1-0-133
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  + 2.49·2-s − 1.49·3-s + 4.20·4-s − 3.71·6-s + 5.49·8-s − 0.777·9-s − 3.42·11-s − 6.26·12-s − 6.26·13-s + 5.26·16-s − 5.20·17-s − 1.93·18-s − 2.22·19-s − 8.53·22-s + 2.91·23-s − 8.18·24-s − 5·25-s − 15.6·26-s + 5.63·27-s + 4.98·29-s − 0.854·31-s + 2.14·32-s + 5.10·33-s − 12.9·34-s − 3.26·36-s + 0.268·37-s − 5.53·38-s + ⋯
L(s)  = 1  + 1.76·2-s − 0.860·3-s + 2.10·4-s − 1.51·6-s + 1.94·8-s − 0.259·9-s − 1.03·11-s − 1.80·12-s − 1.73·13-s + 1.31·16-s − 1.26·17-s − 0.456·18-s − 0.509·19-s − 1.81·22-s + 0.608·23-s − 1.67·24-s − 25-s − 3.06·26-s + 1.08·27-s + 0.925·29-s − 0.153·31-s + 0.378·32-s + 0.889·33-s − 2.22·34-s − 0.544·36-s + 0.0440·37-s − 0.898·38-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: \(1\)
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 \( 1 - 2.49T + 2T^{2} \)
3 \( 1 + 1.49T + 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 3.42T + 11T^{2} \)
13 \( 1 + 6.26T + 13T^{2} \)
17 \( 1 + 5.20T + 17T^{2} \)
19 \( 1 + 2.22T + 19T^{2} \)
23 \( 1 - 2.91T + 23T^{2} \)
29 \( 1 - 4.98T + 29T^{2} \)
31 \( 1 + 0.854T + 31T^{2} \)
37 \( 1 - 0.268T + 37T^{2} \)
43 \( 1 + 3.20T + 43T^{2} \)
47 \( 1 - 4.14T + 47T^{2} \)
53 \( 1 + 3.42T + 53T^{2} \)
59 \( 1 + 2.12T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 - 6.12T + 67T^{2} \)
71 \( 1 + 7.55T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 6.26T + 89T^{2} \)
97 \( 1 + 8.06T + 97T^{2} \)
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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

−8.630340393445771560208434823157, −7.55558071237669283913842001020, −6.81904310227978894492038661843, −6.16486887556350090394728204908, −5.23346997891074903716521334247, −4.94952083500728255524984265889, −4.10720660670331164054673183032, −2.78306489845453934112263348417, −2.27594171943160421296334322268, 0, 2.27594171943160421296334322268, 2.78306489845453934112263348417, 4.10720660670331164054673183032, 4.94952083500728255524984265889, 5.23346997891074903716521334247, 6.16486887556350090394728204908, 6.81904310227978894492038661843, 7.55558071237669283913842001020, 8.630340393445771560208434823157

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