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Properties

Label 1-2009-2009.323-r0-0-0
Degree $1$
Conductor $2009$
Sign $0.0182 - 0.999i$
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  + (−0.936 + 0.351i)2-s + (0.433 + 0.900i)3-s + (0.753 − 0.657i)4-s + (−0.983 − 0.178i)5-s + (−0.722 − 0.691i)6-s + (−0.473 + 0.880i)8-s + (−0.623 + 0.781i)9-s + (0.983 − 0.178i)10-s + (−0.266 + 0.963i)11-s + (0.919 + 0.393i)12-s + (−0.351 − 0.936i)13-s + (−0.266 − 0.963i)15-s + (0.134 − 0.990i)16-s + (−0.657 + 0.753i)17-s + (0.309 − 0.951i)18-s + (−0.951 + 0.309i)19-s + ⋯
L(s)  = 1  + (−0.936 + 0.351i)2-s + (0.433 + 0.900i)3-s + (0.753 − 0.657i)4-s + (−0.983 − 0.178i)5-s + (−0.722 − 0.691i)6-s + (−0.473 + 0.880i)8-s + (−0.623 + 0.781i)9-s + (0.983 − 0.178i)10-s + (−0.266 + 0.963i)11-s + (0.919 + 0.393i)12-s + (−0.351 − 0.936i)13-s + (−0.266 − 0.963i)15-s + (0.134 − 0.990i)16-s + (−0.657 + 0.753i)17-s + (0.309 − 0.951i)18-s + (−0.951 + 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0182 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0182 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.0182 - 0.999i$
Analytic conductor: \(9.32975\)
Root analytic conductor: \(9.32975\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2009,\ (0:\ ),\ 0.0182 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06643399824 + 0.06523324606i\)
\(L(\frac12)\) \(\approx\) \(-0.06643399824 + 0.06523324606i\)
\(L(1)\) \(\approx\) \(0.4543193722 + 0.2783558350i\)
\(L(1)\) \(\approx\) \(0.4543193722 + 0.2783558350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.936 + 0.351i)T \)
3 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (-0.983 - 0.178i)T \)
11 \( 1 + (-0.266 + 0.963i)T \)
13 \( 1 + (-0.351 - 0.936i)T \)
17 \( 1 + (-0.657 + 0.753i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (0.858 + 0.512i)T \)
29 \( 1 + (0.919 + 0.393i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.393 + 0.919i)T \)
43 \( 1 + (0.691 - 0.722i)T \)
47 \( 1 + (0.351 + 0.936i)T \)
53 \( 1 + (-0.657 - 0.753i)T \)
59 \( 1 + (-0.691 + 0.722i)T \)
61 \( 1 + (-0.858 + 0.512i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.919 + 0.393i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.351 - 0.936i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−19.25939968672429145879636097268, −18.83371600028225745804723743340, −18.25031533865676555964379330991, −17.3369144073279038976323636538, −16.61273353553492559632689889297, −15.883180904161784931209235124301, −15.10790042204279029445202037902, −14.2706317868738002920134227481, −13.36675908541919231882649792230, −12.559122317069146450112560102208, −11.93514700758718280627795794934, −11.14849376751876162741924576451, −10.79408556345826391566158842244, −9.3336541590794793094278800680, −8.84794744315617668940570654665, −8.190106449560848884659707734909, −7.42001533617979588940203036698, −6.837593353159351458433778267404, −6.15828936135134613125868973328, −4.57811461277736051643627107109, −3.61968623541361635673285757680, −2.753743568109433194534426708467, −2.17498550496663802520521249417, −0.893758772790469375376798582607, −0.05037370262957058150682622940, 1.532042941308322747811915757273, 2.613743763263550904541864114965, 3.43101016014027925767254802135, 4.55206357226279045506776398926, 5.07788925668371386714080663564, 6.20660790495178002106275040602, 7.3129711085254800689660067904, 7.794182597263268404037764726283, 8.689214645141210829968053911, 9.03110891676834585789548890619, 10.286019110352514703656364017816, 10.4723582131012751113711835003, 11.3124917527455128166827892352, 12.295509798340835742415066405454, 13.03634094026627066467896913073, 14.452292819412655163484953245955, 14.92370924090628560818341095872, 15.59854748061426522919264613311, 15.833227485735561216322928713640, 17.013916293742155984989941821266, 17.29788732523709252936372546586, 18.29772591270217974103504800428, 19.363655631429448677710746642212, 19.59011587247212267433746343042, 20.374075198084876093224045027877

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