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Properties

Label 1-2009-2009.136-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.994 - 0.101i$
Analytic cond. $9.32975$
Root an. cond. $9.32975$
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  + (0.850 − 0.525i)2-s + (0.884 + 0.467i)3-s + (0.447 − 0.894i)4-s + (0.486 + 0.873i)5-s + (0.997 − 0.0672i)6-s + (−0.0896 − 0.995i)8-s + (0.563 + 0.826i)9-s + (0.873 + 0.486i)10-s + (−0.877 + 0.480i)11-s + (0.813 − 0.581i)12-s + (0.957 − 0.287i)13-s + (0.0224 + 0.999i)15-s + (−0.599 − 0.800i)16-s + (0.748 − 0.663i)17-s + (0.913 + 0.406i)18-s + (0.358 − 0.933i)19-s + ⋯
L(s)  = 1  + (0.850 − 0.525i)2-s + (0.884 + 0.467i)3-s + (0.447 − 0.894i)4-s + (0.486 + 0.873i)5-s + (0.997 − 0.0672i)6-s + (−0.0896 − 0.995i)8-s + (0.563 + 0.826i)9-s + (0.873 + 0.486i)10-s + (−0.877 + 0.480i)11-s + (0.813 − 0.581i)12-s + (0.957 − 0.287i)13-s + (0.0224 + 0.999i)15-s + (−0.599 − 0.800i)16-s + (0.748 − 0.663i)17-s + (0.913 + 0.406i)18-s + (0.358 − 0.933i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.994 - 0.101i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.994 - 0.101i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.454810200 - 0.2258836806i\)
\(L(\frac12)\) \(\approx\) \(4.454810200 - 0.2258836806i\)
\(L(1)\) \(\approx\) \(2.533142844 - 0.1891610587i\)
\(L(1)\) \(\approx\) \(2.533142844 - 0.1891610587i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.850 - 0.525i)T \)
3 \( 1 + (0.884 + 0.467i)T \)
5 \( 1 + (0.486 + 0.873i)T \)
11 \( 1 + (-0.877 + 0.480i)T \)
13 \( 1 + (0.957 - 0.287i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (0.358 - 0.933i)T \)
23 \( 1 + (0.842 + 0.538i)T \)
29 \( 1 + (-0.910 - 0.413i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.163 + 0.986i)T \)
43 \( 1 + (0.657 - 0.753i)T \)
47 \( 1 + (0.973 + 0.229i)T \)
53 \( 1 + (-0.948 + 0.316i)T \)
59 \( 1 + (-0.193 + 0.981i)T \)
61 \( 1 + (-0.460 + 0.887i)T \)
67 \( 1 + (0.544 - 0.838i)T \)
71 \( 1 + (-0.413 - 0.910i)T \)
73 \( 1 + (-0.997 + 0.0747i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.685 - 0.727i)T \)
97 \( 1 + (0.453 + 0.891i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.373031979674377909605529973149, −19.206381144447930013727280505994, −18.546409812743765357532504444234, −17.70142658277245065713050145250, −16.86304564584508072139911622933, −16.09540870389035921161351664686, −15.65939790438598562761885635994, −14.51768577852973679939161825205, −14.13333288708542521788283215276, −13.363702860354894734794947718232, −12.7385165751710210695068509463, −12.41676147576386531357902024992, −11.24542838555883942733946131249, −10.26992241332525879229307446686, −9.165635078600804532609053080184, −8.47520243876159956213446108096, −8.001910485926325661730312995552, −7.1527108985759741554701773845, −6.08812531458561260263092381531, −5.643690571223543924054665993675, −4.62738700205188131607007726396, −3.70200571375935975769174511281, −3.05248043866010835680890472792, −2.00018187235481098080458255227, −1.17905281714674589843051956910, 1.24225006096360792440707081978, 2.3140350052540208673605917766, 2.90598084659614070243473224564, 3.45926472211474596883003802028, 4.488593679455159263703544716385, 5.30158944283764866586879732610, 6.03317461913421817750210608524, 7.20532338320992547569336553328, 7.638179929331699507605587492129, 9.02880704765154792971315728586, 9.67322832529296259551135272959, 10.37478771417998599043597722388, 10.93386094004166731816400442777, 11.70629891879035353158284225857, 12.92227582989727120844430217635, 13.526956221832969208798853979223, 13.847928605241118669282125873898, 14.79652077339291474562840174382, 15.4612435378692493558254982659, 15.67751387126890794858306391569, 16.93683953974595084900894019940, 18.10601235375113381095479075549, 18.7264494545792836418579712130, 19.201717488180660147917981326026, 20.252840001525237800479511165122

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