Properties

Label 2-2009-1.1-c1-0-99
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  − 2.45·2-s + 1.45·3-s + 4.03·4-s + 2.26·5-s − 3.58·6-s − 5.00·8-s − 0.876·9-s − 5.57·10-s − 5.41·11-s + 5.88·12-s − 3.23·13-s + 3.30·15-s + 4.22·16-s − 2.83·17-s + 2.15·18-s + 4.32·19-s + 9.15·20-s + 13.3·22-s + 6.99·23-s − 7.29·24-s + 0.138·25-s + 7.95·26-s − 5.64·27-s + 8.06·29-s − 8.11·30-s − 9.18·31-s − 0.374·32-s + ⋯
L(s)  = 1  − 1.73·2-s + 0.841·3-s + 2.01·4-s + 1.01·5-s − 1.46·6-s − 1.77·8-s − 0.292·9-s − 1.76·10-s − 1.63·11-s + 1.69·12-s − 0.897·13-s + 0.852·15-s + 1.05·16-s − 0.686·17-s + 0.507·18-s + 0.991·19-s + 2.04·20-s + 2.83·22-s + 1.45·23-s − 1.48·24-s + 0.0277·25-s + 1.55·26-s − 1.08·27-s + 1.49·29-s − 1.48·30-s − 1.64·31-s − 0.0662·32-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 \( 1 + 2.45T + 2T^{2} \)
3 \( 1 - 1.45T + 3T^{2} \)
5 \( 1 - 2.26T + 5T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 + 2.83T + 17T^{2} \)
19 \( 1 - 4.32T + 19T^{2} \)
23 \( 1 - 6.99T + 23T^{2} \)
29 \( 1 - 8.06T + 29T^{2} \)
31 \( 1 + 9.18T + 31T^{2} \)
37 \( 1 + 0.0469T + 37T^{2} \)
43 \( 1 + 6.31T + 43T^{2} \)
47 \( 1 + 5.26T + 47T^{2} \)
53 \( 1 - 6.43T + 53T^{2} \)
59 \( 1 + 2.45T + 59T^{2} \)
61 \( 1 + 5.28T + 61T^{2} \)
67 \( 1 + 8.78T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 2.42T + 73T^{2} \)
79 \( 1 - 4.92T + 79T^{2} \)
83 \( 1 - 1.63T + 83T^{2} \)
89 \( 1 + 1.68T + 89T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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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

−8.891971736169585603043261049138, −8.158373907303274556649178513113, −7.49667080212994007533313460810, −6.86994516713706088967838273447, −5.70658465295966520191905091239, −4.95149522931480885821527814426, −2.95092082041346917091136601894, −2.58138860830674377135684017116, −1.61818608331242024158537996988, 0, 1.61818608331242024158537996988, 2.58138860830674377135684017116, 2.95092082041346917091136601894, 4.95149522931480885821527814426, 5.70658465295966520191905091239, 6.86994516713706088967838273447, 7.49667080212994007533313460810, 8.158373907303274556649178513113, 8.891971736169585603043261049138

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