Properties

Label 2-2009-1.1-c1-0-108
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  + 0.618·2-s + 1.61·3-s − 1.61·4-s − 1.61·5-s + 1.00·6-s − 2.23·8-s − 0.381·9-s − 1.00·10-s − 11-s − 2.61·12-s + 6.23·13-s − 2.61·15-s + 1.85·16-s + 4.23·17-s − 0.236·18-s − 2.85·19-s + 2.61·20-s − 0.618·22-s − 3.61·23-s − 3.61·24-s − 2.38·25-s + 3.85·26-s − 5.47·27-s − 5.85·29-s − 1.61·30-s − 8.09·31-s + 5.61·32-s + ⋯
L(s)  = 1  + 0.437·2-s + 0.934·3-s − 0.809·4-s − 0.723·5-s + 0.408·6-s − 0.790·8-s − 0.127·9-s − 0.316·10-s − 0.301·11-s − 0.755·12-s + 1.72·13-s − 0.675·15-s + 0.463·16-s + 1.02·17-s − 0.0556·18-s − 0.654·19-s + 0.585·20-s − 0.131·22-s − 0.754·23-s − 0.738·24-s − 0.476·25-s + 0.755·26-s − 1.05·27-s − 1.08·29-s − 0.295·30-s − 1.45·31-s + 0.993·32-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 \( 1 - 0.618T + 2T^{2} \)
3 \( 1 - 1.61T + 3T^{2} \)
5 \( 1 + 1.61T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 6.23T + 13T^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 + 2.85T + 19T^{2} \)
23 \( 1 + 3.61T + 23T^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
31 \( 1 + 8.09T + 31T^{2} \)
37 \( 1 + 2.76T + 37T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 9.70T + 47T^{2} \)
53 \( 1 - 2.38T + 53T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 + 9.47T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 + 5.94T + 71T^{2} \)
73 \( 1 - 9.94T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 5.47T + 83T^{2} \)
89 \( 1 - 9.32T + 89T^{2} \)
97 \( 1 + 2.56T + 97T^{2} \)
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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

−8.761867100193339664683143410959, −8.030644668258598871134325168454, −7.60138709349562027439852209703, −6.11047321348933896266641556122, −5.60756832659515713103505028019, −4.40828389267130868852530131494, −3.52148982483247484138387134864, −3.37068856664004201484573908035, −1.75407853264927128953859081447, 0, 1.75407853264927128953859081447, 3.37068856664004201484573908035, 3.52148982483247484138387134864, 4.40828389267130868852530131494, 5.60756832659515713103505028019, 6.11047321348933896266641556122, 7.60138709349562027439852209703, 8.030644668258598871134325168454, 8.761867100193339664683143410959

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