Properties

Label 2-2009-1.1-c1-0-35
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  + 1.17·2-s − 2.72·3-s − 0.618·4-s + 2.82·5-s − 3.20·6-s − 3.07·8-s + 4.44·9-s + 3.32·10-s + 4.71·11-s + 1.68·12-s − 2.95·13-s − 7.71·15-s − 2.38·16-s − 5.06·17-s + 5.22·18-s + 4.50·19-s − 1.74·20-s + 5.54·22-s + 1.60·23-s + 8.39·24-s + 3.00·25-s − 3.47·26-s − 3.94·27-s + 7.93·29-s − 9.07·30-s − 7.31·31-s + 3.35·32-s + ⋯
L(s)  = 1  + 0.831·2-s − 1.57·3-s − 0.309·4-s + 1.26·5-s − 1.30·6-s − 1.08·8-s + 1.48·9-s + 1.05·10-s + 1.42·11-s + 0.486·12-s − 0.820·13-s − 1.99·15-s − 0.595·16-s − 1.22·17-s + 1.23·18-s + 1.03·19-s − 0.390·20-s + 1.18·22-s + 0.335·23-s + 1.71·24-s + 0.600·25-s − 0.681·26-s − 0.758·27-s + 1.47·29-s − 1.65·30-s − 1.31·31-s + 0.593·32-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

\(L(1)\) \(\approx\) \(1.602261868\)
\(L(\frac12)\) \(\approx\) \(1.602261868\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.17T + 2T^{2} \)
3 \( 1 + 2.72T + 3T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 - 4.71T + 11T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 - 4.50T + 19T^{2} \)
23 \( 1 - 1.60T + 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + 7.31T + 31T^{2} \)
37 \( 1 - 1.35T + 37T^{2} \)
43 \( 1 + 1.02T + 43T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 - 1.72T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 6.96T + 61T^{2} \)
67 \( 1 - 0.648T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 5.45T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 + 6.86T + 97T^{2} \)
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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

−9.537088766030463593896795666276, −8.584946533443979049481447034905, −6.92462293505147214404464140263, −6.55219317090923682802526728489, −5.83509469996842078700176244681, −5.12434363217047020304484595775, −4.67331023382092660427894972290, −3.59619229762121262240087710528, −2.16567053885917478840314586831, −0.822317597455318377786862549921, 0.822317597455318377786862549921, 2.16567053885917478840314586831, 3.59619229762121262240087710528, 4.67331023382092660427894972290, 5.12434363217047020304484595775, 5.83509469996842078700176244681, 6.55219317090923682802526728489, 6.92462293505147214404464140263, 8.584946533443979049481447034905, 9.537088766030463593896795666276

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