Properties

Label 2-2009-1.1-c1-0-6
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  + 1.44·2-s − 1.55·3-s + 0.0881·4-s − 3.60·5-s − 2.24·6-s − 2.76·8-s − 0.582·9-s − 5.20·10-s − 4.49·11-s − 0.137·12-s − 4.13·13-s + 5.60·15-s − 4.16·16-s − 2.02·17-s − 0.841·18-s + 8.29·19-s − 0.317·20-s − 6.49·22-s − 1.06·23-s + 4.29·24-s + 7.98·25-s − 5.97·26-s + 5.57·27-s + 6.98·29-s + 8.09·30-s − 9.48·31-s − 0.498·32-s + ⋯
L(s)  = 1  + 1.02·2-s − 0.897·3-s + 0.0440·4-s − 1.61·5-s − 0.917·6-s − 0.976·8-s − 0.194·9-s − 1.64·10-s − 1.35·11-s − 0.0395·12-s − 1.14·13-s + 1.44·15-s − 1.04·16-s − 0.490·17-s − 0.198·18-s + 1.90·19-s − 0.0710·20-s − 1.38·22-s − 0.221·23-s + 0.876·24-s + 1.59·25-s − 1.17·26-s + 1.07·27-s + 1.29·29-s + 1.47·30-s − 1.70·31-s − 0.0880·32-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

\(L(1)\) \(\approx\) \(0.4167471135\)
\(L(\frac12)\) \(\approx\) \(0.4167471135\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.44T + 2T^{2} \)
3 \( 1 + 1.55T + 3T^{2} \)
5 \( 1 + 3.60T + 5T^{2} \)
11 \( 1 + 4.49T + 11T^{2} \)
13 \( 1 + 4.13T + 13T^{2} \)
17 \( 1 + 2.02T + 17T^{2} \)
19 \( 1 - 8.29T + 19T^{2} \)
23 \( 1 + 1.06T + 23T^{2} \)
29 \( 1 - 6.98T + 29T^{2} \)
31 \( 1 + 9.48T + 31T^{2} \)
37 \( 1 + 3.74T + 37T^{2} \)
43 \( 1 + 0.515T + 43T^{2} \)
47 \( 1 + 2.75T + 47T^{2} \)
53 \( 1 + 3.50T + 53T^{2} \)
59 \( 1 - 0.933T + 59T^{2} \)
61 \( 1 - 6.59T + 61T^{2} \)
67 \( 1 + 15.3T + 67T^{2} \)
71 \( 1 - 5.20T + 71T^{2} \)
73 \( 1 - 0.219T + 73T^{2} \)
79 \( 1 - 6.21T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + 0.841T + 97T^{2} \)
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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

−9.089436042997955140785357843394, −8.154061920608499167104713969903, −7.48438010794385192673065596103, −6.74468152094969572763853178761, −5.51140958432480097000922858521, −5.11794426808117334530315993013, −4.49114217773277642118662473924, −3.41623856830810583787432527366, −2.76469789548241240077393575191, −0.36101717244637400149124311412, 0.36101717244637400149124311412, 2.76469789548241240077393575191, 3.41623856830810583787432527366, 4.49114217773277642118662473924, 5.11794426808117334530315993013, 5.51140958432480097000922858521, 6.74468152094969572763853178761, 7.48438010794385192673065596103, 8.154061920608499167104713969903, 9.089436042997955140785357843394

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