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Properties

Label 1-6017-6017.2009-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.883 + 0.468i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (0.0379 + 0.999i)2-s + (0.963 − 0.266i)3-s + (−0.997 + 0.0758i)4-s + (0.492 + 0.870i)5-s + (0.302 + 0.953i)6-s + (0.897 − 0.440i)7-s + (−0.113 − 0.993i)8-s + (0.858 − 0.512i)9-s + (−0.851 + 0.524i)10-s + (−0.940 + 0.338i)12-s + (−0.134 + 0.990i)13-s + (0.473 + 0.880i)14-s + (0.705 + 0.708i)15-s + (0.988 − 0.151i)16-s + (−0.923 + 0.383i)17-s + (0.545 + 0.838i)18-s + ⋯
L(s)  = 1  + (0.0379 + 0.999i)2-s + (0.963 − 0.266i)3-s + (−0.997 + 0.0758i)4-s + (0.492 + 0.870i)5-s + (0.302 + 0.953i)6-s + (0.897 − 0.440i)7-s + (−0.113 − 0.993i)8-s + (0.858 − 0.512i)9-s + (−0.851 + 0.524i)10-s + (−0.940 + 0.338i)12-s + (−0.134 + 0.990i)13-s + (0.473 + 0.880i)14-s + (0.705 + 0.708i)15-s + (0.988 − 0.151i)16-s + (−0.923 + 0.383i)17-s + (0.545 + 0.838i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.883 + 0.468i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (2009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.883 + 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6611333446 + 2.655912076i\)
\(L(\frac12)\) \(\approx\) \(0.6611333446 + 2.655912076i\)
\(L(1)\) \(\approx\) \(1.229253181 + 0.9801489493i\)
\(L(1)\) \(\approx\) \(1.229253181 + 0.9801489493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.0379 + 0.999i)T \)
3 \( 1 + (0.963 - 0.266i)T \)
5 \( 1 + (0.492 + 0.870i)T \)
7 \( 1 + (0.897 - 0.440i)T \)
13 \( 1 + (-0.134 + 0.990i)T \)
17 \( 1 + (-0.923 + 0.383i)T \)
19 \( 1 + (0.249 + 0.968i)T \)
23 \( 1 + (-0.725 + 0.688i)T \)
29 \( 1 + (0.720 - 0.693i)T \)
31 \( 1 + (0.993 + 0.117i)T \)
37 \( 1 + (-0.348 + 0.937i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.700 + 0.713i)T \)
47 \( 1 + (-0.262 + 0.964i)T \)
53 \( 1 + (0.891 - 0.452i)T \)
59 \( 1 + (-0.644 + 0.764i)T \)
61 \( 1 + (0.686 + 0.727i)T \)
67 \( 1 + (0.940 - 0.338i)T \)
71 \( 1 + (-0.945 - 0.325i)T \)
73 \( 1 + (0.455 + 0.890i)T \)
79 \( 1 + (-0.894 - 0.446i)T \)
83 \( 1 + (0.958 - 0.285i)T \)
89 \( 1 + (0.479 + 0.877i)T \)
97 \( 1 + (-0.341 - 0.939i)T \)
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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

−17.74991053546441212604789389675, −16.98738341033894377238378176029, −15.943065944334624071655900897299, −15.38594644648015176762575121597, −14.65917561372144812890347554401, −13.932782734033488791285942346651, −13.50688776289434652504851918470, −12.88098754788060945689457986744, −12.21580897058165303107737234168, −11.57239753090753488805039455523, −10.633670599649671541557705952790, −10.167921686380548310515362051623, −9.41312615190309706149523175458, −8.73041232515116002900794096708, −8.45078796438748253572668811280, −7.803688408778335194805324610894, −6.64982250013601356261058558020, −5.41732578225846793378316096125, −4.93836500780125415305506593872, −4.4852518718619690374088489201, −3.58635190199900607021147369502, −2.60584014575487303811425931648, −2.26292127503701988973813855908, −1.463388791635804557487008229123, −0.56057612468743425175689760585, 1.29972877124225146889426852231, 1.86908329944245652739370612987, 2.807644957452517770959529550586, 3.78223860797918359264983316975, 4.24433076389694383089487723572, 5.06052717824625583325997812469, 6.13742320972841859486732287630, 6.62061787346729167755444738375, 7.232182357378563124503923128625, 7.945667907670513308098871331840, 8.39615446593376469207552189933, 9.169255063203710270651649670033, 10.02690066401110570411143953844, 10.25887833559072669707680116138, 11.5127463202766647218557714083, 12.09891818664440189165392032062, 13.26992959544312132045635430177, 13.69512567370565845634675003510, 14.0918473976799301228723706188, 14.64904734366989949423578705682, 15.220771444033873448402311108281, 15.80475404635859912233657202241, 16.724357570444861894492088621540, 17.491016896768260008333515090861, 17.86284778490854869711269547057

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