Properties

Label 2-2009-1.1-c1-0-56
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  − 0.460·2-s − 0.539·3-s − 1.78·4-s − 4.10·5-s + 0.248·6-s + 1.74·8-s − 2.70·9-s + 1.88·10-s + 2.76·11-s + 0.965·12-s + 4.05·13-s + 2.21·15-s + 2.77·16-s − 5.22·17-s + 1.24·18-s + 0.109·19-s + 7.33·20-s − 1.27·22-s + 6.08·23-s − 0.941·24-s + 11.8·25-s − 1.86·26-s + 3.08·27-s + 2.25·29-s − 1.01·30-s + 1.18·31-s − 4.76·32-s + ⋯
L(s)  = 1  − 0.325·2-s − 0.311·3-s − 0.894·4-s − 1.83·5-s + 0.101·6-s + 0.616·8-s − 0.902·9-s + 0.597·10-s + 0.834·11-s + 0.278·12-s + 1.12·13-s + 0.571·15-s + 0.693·16-s − 1.26·17-s + 0.293·18-s + 0.0250·19-s + 1.63·20-s − 0.271·22-s + 1.26·23-s − 0.192·24-s + 2.36·25-s − 0.366·26-s + 0.592·27-s + 0.418·29-s − 0.186·30-s + 0.212·31-s − 0.842·32-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 \( 1 + 0.460T + 2T^{2} \)
3 \( 1 + 0.539T + 3T^{2} \)
5 \( 1 + 4.10T + 5T^{2} \)
11 \( 1 - 2.76T + 11T^{2} \)
13 \( 1 - 4.05T + 13T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 - 0.109T + 19T^{2} \)
23 \( 1 - 6.08T + 23T^{2} \)
29 \( 1 - 2.25T + 29T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
43 \( 1 + 7.93T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 6.23T + 53T^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 + 1.35T + 67T^{2} \)
71 \( 1 + 7.90T + 71T^{2} \)
73 \( 1 + 9.88T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 0.852T + 89T^{2} \)
97 \( 1 + 9.92T + 97T^{2} \)
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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

−8.695935683602694694538501102734, −8.325020777682234876030365011986, −7.25037890503294240867123163741, −6.59726751178773381521546329118, −5.40908198310232626954349656470, −4.49966951751074924821877384455, −3.90201972871781149247036894347, −3.10596952704629512806139718781, −1.06648634595800888515613576231, 0, 1.06648634595800888515613576231, 3.10596952704629512806139718781, 3.90201972871781149247036894347, 4.49966951751074924821877384455, 5.40908198310232626954349656470, 6.59726751178773381521546329118, 7.25037890503294240867123163741, 8.325020777682234876030365011986, 8.695935683602694694538501102734

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