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Properties

Label 1-6027-6027.269-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.982 + 0.186i$
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  + (0.337 − 0.941i)2-s + (−0.772 − 0.635i)4-s + (0.575 − 0.817i)5-s + (−0.858 + 0.512i)8-s + (−0.575 − 0.817i)10-s + (−0.925 + 0.379i)11-s + (0.983 − 0.178i)13-s + (0.193 + 0.981i)16-s + (0.163 + 0.986i)17-s + (0.913 − 0.406i)19-s + (−0.963 + 0.266i)20-s + (0.0448 + 0.998i)22-s + (−0.712 + 0.701i)23-s + (−0.337 − 0.941i)25-s + (0.163 − 0.986i)26-s + ⋯
L(s)  = 1  + (0.337 − 0.941i)2-s + (−0.772 − 0.635i)4-s + (0.575 − 0.817i)5-s + (−0.858 + 0.512i)8-s + (−0.575 − 0.817i)10-s + (−0.925 + 0.379i)11-s + (0.983 − 0.178i)13-s + (0.193 + 0.981i)16-s + (0.163 + 0.986i)17-s + (0.913 − 0.406i)19-s + (−0.963 + 0.266i)20-s + (0.0448 + 0.998i)22-s + (−0.712 + 0.701i)23-s + (−0.337 − 0.941i)25-s + (0.163 − 0.986i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.982 + 0.186i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.982 + 0.186i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1360117421 - 1.442317843i\)
\(L(\frac12)\) \(\approx\) \(-0.1360117421 - 1.442317843i\)
\(L(1)\) \(\approx\) \(0.8581735696 - 0.7535555276i\)
\(L(1)\) \(\approx\) \(0.8581735696 - 0.7535555276i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.337 - 0.941i)T \)
5 \( 1 + (0.575 - 0.817i)T \)
11 \( 1 + (-0.925 + 0.379i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (0.163 + 0.986i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.712 + 0.701i)T \)
29 \( 1 + (-0.550 - 0.834i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.998 - 0.0598i)T \)
43 \( 1 + (-0.393 - 0.919i)T \)
47 \( 1 + (0.337 - 0.941i)T \)
53 \( 1 + (-0.772 - 0.635i)T \)
59 \( 1 + (-0.599 + 0.800i)T \)
61 \( 1 + (-0.251 + 0.967i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.646 - 0.762i)T \)
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

−18.190262132031168851371408766380, −17.40645603794266769024088991773, −16.4752927509594116310115427929, −16.12702715105665642205419029406, −15.526381699819553552465149462615, −14.65835717143244605552430383584, −14.168044920758204613325540158669, −13.65845012550970849918649440804, −13.066337642292188877462570415335, −12.36645928292191023109114247704, −11.30621063112423480064753737502, −10.924898431971279829453953713647, −9.82904646231387618452208931686, −9.473977644507251527899771760180, −8.57344103116154005255459597979, −7.72780669752931376776759286888, −7.41730912429618664887014054323, −6.38878414959717887858769004725, −6.010391694695776875556553808623, −5.34186544006603093071864942586, −4.59846268816542559946717497495, −3.55426062723004937505467183290, −3.09525204973113676158207040843, −2.26236215970110910108152842120, −1.035761638282204910372317945125, 0.34092657245727975644903679962, 1.41394668097129438926679720262, 1.85087203062520844411449963560, 2.76247822667360084456847253056, 3.62033954591974466892285835121, 4.255181156347839157038327882710, 5.111956969278839696679014194739, 5.66563115510352407588354543987, 6.11603730868239483815262937464, 7.40685145168749979178110421129, 8.23469871064310935232965152519, 8.795912366672002405121901486970, 9.555688643735819625801834575865, 10.11843847343828499862142664840, 10.68892673448608386965763320142, 11.5043979120699455115703472245, 12.131501467577422684405335133969, 12.88144445746592473533079058199, 13.31391721338511375534825378040, 13.72745608708091006023395595519, 14.59147213692798750924529843799, 15.419415513761129668641528893847, 15.92946699427977244307995095959, 16.81497155941071944151615561257, 17.59742600698149932941358204638

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