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Properties

Label 1-2009-2009.272-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.981 - 0.192i$
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  + (0.834 − 0.550i)2-s + (−0.943 + 0.330i)3-s + (0.393 − 0.919i)4-s + (−0.880 + 0.473i)5-s + (−0.605 + 0.795i)6-s + (−0.178 − 0.983i)8-s + (0.781 − 0.623i)9-s + (−0.473 + 0.880i)10-s + (0.674 + 0.738i)11-s + (−0.0672 + 0.997i)12-s + (−0.979 − 0.200i)13-s + (0.674 − 0.738i)15-s + (−0.691 − 0.722i)16-s + (0.372 − 0.928i)17-s + (0.309 − 0.951i)18-s + (−0.453 + 0.891i)19-s + ⋯
L(s)  = 1  + (0.834 − 0.550i)2-s + (−0.943 + 0.330i)3-s + (0.393 − 0.919i)4-s + (−0.880 + 0.473i)5-s + (−0.605 + 0.795i)6-s + (−0.178 − 0.983i)8-s + (0.781 − 0.623i)9-s + (−0.473 + 0.880i)10-s + (0.674 + 0.738i)11-s + (−0.0672 + 0.997i)12-s + (−0.979 − 0.200i)13-s + (0.674 − 0.738i)15-s + (−0.691 − 0.722i)16-s + (0.372 − 0.928i)17-s + (0.309 − 0.951i)18-s + (−0.453 + 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.981 - 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417931237 - 0.1375641538i\)
\(L(\frac12)\) \(\approx\) \(1.417931237 - 0.1375641538i\)
\(L(1)\) \(\approx\) \(1.063038263 - 0.1925054427i\)
\(L(1)\) \(\approx\) \(1.063038263 - 0.1925054427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.834 - 0.550i)T \)
3 \( 1 + (-0.943 + 0.330i)T \)
5 \( 1 + (-0.880 + 0.473i)T \)
11 \( 1 + (0.674 + 0.738i)T \)
13 \( 1 + (-0.979 - 0.200i)T \)
17 \( 1 + (0.372 - 0.928i)T \)
19 \( 1 + (-0.453 + 0.891i)T \)
23 \( 1 + (0.995 + 0.0896i)T \)
29 \( 1 + (-0.0672 + 0.997i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.753 - 0.657i)T \)
43 \( 1 + (-0.990 + 0.134i)T \)
47 \( 1 + (0.200 - 0.979i)T \)
53 \( 1 + (0.372 + 0.928i)T \)
59 \( 1 + (-0.134 - 0.990i)T \)
61 \( 1 + (0.0896 + 0.995i)T \)
67 \( 1 + (0.156 + 0.987i)T \)
71 \( 1 + (-0.997 + 0.0672i)T \)
73 \( 1 + (0.781 - 0.623i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.200 + 0.979i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−19.79664370917351734362706326050, −19.36932527496187269873522980329, −18.50281891638296186427649724239, −17.29128851544571316750731154883, −16.86921205963449158788282728493, −16.57089873925921655819009046440, −15.45892638653333071873694547507, −15.06609305210832813666312557907, −14.126059990110802213497769676115, −12.96336722345819956579111692761, −12.86333861465056749207261889064, −11.76888846226576609008878077574, −11.515593364853673432399125466, −10.72522862407723128800511233482, −9.39702560992568158848130137460, −8.41759854779436313584900919613, −7.717007233591435197957697080931, −6.966972667900028178877225394425, −6.29761360060030971777295594647, −5.45861057012403006325618669427, −4.67103378083022088863138859389, −4.12583911010059621911186685187, −3.14106112002956946666130473726, −1.91232334633108310998573918974, −0.60885263317891582481091654223, 0.762659174761238332844867948064, 1.899735967117732775725925181404, 3.09877522732406831188976141149, 3.78122880142479540833639262903, 4.64083230958970127757079641162, 5.13509722547618261499621055000, 6.119255025870365078459170630142, 7.16885809928207076095606785798, 7.24261017070555051537924760118, 8.98780542622681288998586039412, 9.86761041893240219869091430288, 10.465342074665894540350008286259, 11.20339128779949928022668065689, 11.89523973222951722817180295395, 12.32397753132573946852695122065, 12.962136223250122172800348091118, 14.32266239822410655922174426135, 14.80445589458659235963891011817, 15.299495276867756867788608790138, 16.30479556146525314246984724837, 16.73859913232259076558007126719, 17.91200657076449470305751461168, 18.5349837980679143634938815903, 19.31150779029917458927018855364, 20.0527569787254710697794491416

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