Properties

Label 2-2009-1.1-c1-0-15
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  + 0.463·2-s + 0.981·3-s − 1.78·4-s − 3.31·5-s + 0.454·6-s − 1.75·8-s − 2.03·9-s − 1.53·10-s − 2.89·11-s − 1.75·12-s − 6.95·13-s − 3.25·15-s + 2.75·16-s + 7.66·17-s − 0.943·18-s + 1.90·19-s + 5.92·20-s − 1.34·22-s + 3.76·23-s − 1.72·24-s + 5.99·25-s − 3.22·26-s − 4.94·27-s − 3.58·29-s − 1.50·30-s + 8.42·31-s + 4.78·32-s + ⋯
L(s)  = 1  + 0.327·2-s + 0.566·3-s − 0.892·4-s − 1.48·5-s + 0.185·6-s − 0.620·8-s − 0.678·9-s − 0.485·10-s − 0.873·11-s − 0.505·12-s − 1.92·13-s − 0.840·15-s + 0.689·16-s + 1.86·17-s − 0.222·18-s + 0.436·19-s + 1.32·20-s − 0.286·22-s + 0.785·23-s − 0.351·24-s + 1.19·25-s − 0.631·26-s − 0.951·27-s − 0.666·29-s − 0.275·30-s + 1.51·31-s + 0.845·32-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

\(L(1)\) \(\approx\) \(0.8471436941\)
\(L(\frac12)\) \(\approx\) \(0.8471436941\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.463T + 2T^{2} \)
3 \( 1 - 0.981T + 3T^{2} \)
5 \( 1 + 3.31T + 5T^{2} \)
11 \( 1 + 2.89T + 11T^{2} \)
13 \( 1 + 6.95T + 13T^{2} \)
17 \( 1 - 7.66T + 17T^{2} \)
19 \( 1 - 1.90T + 19T^{2} \)
23 \( 1 - 3.76T + 23T^{2} \)
29 \( 1 + 3.58T + 29T^{2} \)
31 \( 1 - 8.42T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 - 8.86T + 47T^{2} \)
53 \( 1 - 4.12T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 8.56T + 61T^{2} \)
67 \( 1 + 4.56T + 67T^{2} \)
71 \( 1 - 2.16T + 71T^{2} \)
73 \( 1 - 3.80T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 7.65T + 89T^{2} \)
97 \( 1 + 0.742T + 97T^{2} \)
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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

−9.089570447753722283698765580948, −8.123208092416786016899726755679, −7.83205471079245506332622570424, −7.17810941046703915975292610178, −5.56051651273397208141599301256, −5.08376429194428681802489883026, −4.19809357637507020721546444573, −3.24108960552253890814116314602, −2.77061357522761651835471432131, −0.54568538641980600916578717235, 0.54568538641980600916578717235, 2.77061357522761651835471432131, 3.24108960552253890814116314602, 4.19809357637507020721546444573, 5.08376429194428681802489883026, 5.56051651273397208141599301256, 7.17810941046703915975292610178, 7.83205471079245506332622570424, 8.123208092416786016899726755679, 9.089570447753722283698765580948

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