Properties

Label 1-6027-6027.1412-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.999 + 0.0408i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (0.646 + 0.762i)2-s + (−0.163 + 0.986i)4-s + (0.420 + 0.907i)5-s + (−0.858 + 0.512i)8-s + (−0.420 + 0.907i)10-s + (−0.791 − 0.611i)11-s + (−0.983 + 0.178i)13-s + (−0.946 − 0.323i)16-s + (−0.772 + 0.635i)17-s + (0.104 − 0.994i)19-s + (−0.963 + 0.266i)20-s + (−0.0448 − 0.998i)22-s + (−0.251 − 0.967i)23-s + (−0.646 + 0.762i)25-s + (−0.772 − 0.635i)26-s + ⋯
L(s)  = 1  + (0.646 + 0.762i)2-s + (−0.163 + 0.986i)4-s + (0.420 + 0.907i)5-s + (−0.858 + 0.512i)8-s + (−0.420 + 0.907i)10-s + (−0.791 − 0.611i)11-s + (−0.983 + 0.178i)13-s + (−0.946 − 0.323i)16-s + (−0.772 + 0.635i)17-s + (0.104 − 0.994i)19-s + (−0.963 + 0.266i)20-s + (−0.0448 − 0.998i)22-s + (−0.251 − 0.967i)23-s + (−0.646 + 0.762i)25-s + (−0.772 − 0.635i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.999 + 0.0408i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.999 + 0.0408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.111873826 + 0.02273131854i\)
\(L(\frac12)\) \(\approx\) \(1.111873826 + 0.02273131854i\)
\(L(1)\) \(\approx\) \(0.9859724944 + 0.6207186217i\)
\(L(1)\) \(\approx\) \(0.9859724944 + 0.6207186217i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.646 + 0.762i)T \)
5 \( 1 + (0.420 + 0.907i)T \)
11 \( 1 + (-0.791 - 0.611i)T \)
13 \( 1 + (-0.983 + 0.178i)T \)
17 \( 1 + (-0.772 + 0.635i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.251 - 0.967i)T \)
29 \( 1 + (0.550 + 0.834i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.447 + 0.894i)T \)
43 \( 1 + (-0.393 - 0.919i)T \)
47 \( 1 + (-0.646 - 0.762i)T \)
53 \( 1 + (0.163 - 0.986i)T \)
59 \( 1 + (0.992 + 0.119i)T \)
61 \( 1 + (-0.712 - 0.701i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.550 - 0.834i)T \)
73 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.337 - 0.941i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−17.76453553013020858211818944503, −17.22773254082731528597358292355, −16.22637607208397014255822904565, −15.63977046837554665870717744492, −15.08116676449093754906112379585, −14.16004600701510744884300546642, −13.68133063754203784535753351779, −13.03312224382367150116891542899, −12.43385736021448393083251882192, −11.98431971443958758678346235275, −11.2467375980513303974443464734, −10.318223526729943018535563976606, −9.73581248394243002347513921336, −9.468399478409729629222672284881, −8.40136922827026343248861419316, −7.71091400851265669272141724001, −6.79776556521916345729266944923, −5.842794642379981096077765123560, −5.381941182185870148517924265668, −4.5911069454745107810498365093, −4.27605911355652493059236190876, −3.086390503040325178833973486316, −2.37491193547132626351389082360, −1.79944573289279536707481895396, −0.85905527552764765872189898319, 0.2383086581552500045575919386, 1.98273003811991796314988381451, 2.68544901870386204129269858769, 3.149642474951574077286949231043, 4.12816345190364335640702362911, 4.968430481227081600709157287002, 5.40173888240648711468017308809, 6.54089675224422265942896632670, 6.63919955334651508020114154103, 7.414101391226702035621958206789, 8.362742012625698143116854321473, 8.70964101396203640208691907306, 9.84476289657041328923399799239, 10.40723235860759300242560253225, 11.19931802628620175561670465796, 11.83125264089257042304067135848, 12.69241009056211522938471222245, 13.28513368907215512331897020480, 13.880461265712693714698549312728, 14.419832703963329555953702020185, 15.1422895622572853310193424345, 15.551240385389405344337600864494, 16.30475922117336377837150727638, 17.087166572739901304286134495241, 17.58849726415961403331831980339

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