Properties

Label 2-2009-287.40-c0-0-3
Degree $2$
Conductor $2009$
Sign $0.0633 + 0.997i$
Analytic cond. $1.00262$
Root an. cond. $1.00130$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 − 1.56i)2-s + (0.623 + 1.07i)3-s + (−1.12 − 1.94i)4-s + 2.24·6-s − 2.24·8-s + (−0.277 + 0.480i)9-s + (1.40 − 2.42i)12-s + 1.80·13-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s + (0.5 + 0.866i)18-s + (−0.222 + 0.385i)19-s + (−0.623 + 1.07i)23-s + (−1.40 − 2.42i)24-s + (−0.5 − 0.866i)25-s + (1.62 − 2.81i)26-s + ⋯
L(s)  = 1  + (0.900 − 1.56i)2-s + (0.623 + 1.07i)3-s + (−1.12 − 1.94i)4-s + 2.24·6-s − 2.24·8-s + (−0.277 + 0.480i)9-s + (1.40 − 2.42i)12-s + 1.80·13-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s + (0.5 + 0.866i)18-s + (−0.222 + 0.385i)19-s + (−0.623 + 1.07i)23-s + (−1.40 − 2.42i)24-s + (−0.5 − 0.866i)25-s + (1.62 − 2.81i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(1.00262\)
Root analytic conductor: \(1.00130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2009,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.189096510\)
\(L(\frac12)\) \(\approx\) \(2.189096510\)
\(L(1)\) 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 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.80T + T^{2} \)
17 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.24T + T^{2} \)
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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

−9.501699021753811911276129019053, −8.794163293411639487801782865412, −7.959681809300946824120904898981, −6.35226389030292845089428189104, −5.62379910538386516296562473044, −4.63375378762089689495879156420, −3.87345243392989194229338668329, −3.54652072770348459017211618959, −2.53557174314562538911207541897, −1.39791643264655361206048294006, 1.65919010299767152506857707676, 3.07571694836446014133026789368, 3.90542753497502932814445841013, 4.82836953762482163783993782437, 5.88521498722584552709073820709, 6.49878255232474630489208330425, 6.95447597875377976741427818740, 7.924379939417520550759319958347, 8.416627160696390653759239728244, 8.775038965134369957015150194482

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