Properties

Label 2-2009-1.1-c1-0-126
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  + 1.61·2-s + 3-s + 0.618·4-s − 5-s + 1.61·6-s − 2.23·8-s − 2·9-s − 1.61·10-s + 11-s + 0.618·12-s − 4.47·13-s − 15-s − 4.85·16-s + 4.23·17-s − 3.23·18-s + 1.47·19-s − 0.618·20-s + 1.61·22-s − 8.70·23-s − 2.23·24-s − 4·25-s − 7.23·26-s − 5·27-s + 4.47·29-s − 1.61·30-s − 0.236·31-s − 3.38·32-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.447·5-s + 0.660·6-s − 0.790·8-s − 0.666·9-s − 0.511·10-s + 0.301·11-s + 0.178·12-s − 1.24·13-s − 0.258·15-s − 1.21·16-s + 1.02·17-s − 0.762·18-s + 0.337·19-s − 0.138·20-s + 0.344·22-s − 1.81·23-s − 0.456·24-s − 0.800·25-s − 1.41·26-s − 0.962·27-s + 0.830·29-s − 0.295·30-s − 0.0423·31-s − 0.597·32-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: \(1\)
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 \( 1 - 1.61T + 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 + 8.70T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 0.236T + 31T^{2} \)
37 \( 1 + 3.47T + 37T^{2} \)
43 \( 1 + 6.47T + 43T^{2} \)
47 \( 1 + 3.47T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 9.47T + 61T^{2} \)
67 \( 1 - 5.94T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 2.52T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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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

−8.567918953136780633484257647889, −8.041921234879686164509206930548, −7.17627026258232825692998879268, −6.12306502516773626490928137484, −5.44690620456728997159752121778, −4.60921837020544138460621471676, −3.71531160719648925625651121301, −3.11039342214413083653250917454, −2.10972816409973249933079616325, 0, 2.10972816409973249933079616325, 3.11039342214413083653250917454, 3.71531160719648925625651121301, 4.60921837020544138460621471676, 5.44690620456728997159752121778, 6.12306502516773626490928137484, 7.17627026258232825692998879268, 8.041921234879686164509206930548, 8.567918953136780633484257647889

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