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Properties

Label 1-6027-6027.845-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.971 - 0.237i$
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  + (0.842 − 0.538i)2-s + (0.420 − 0.907i)4-s + (−0.280 + 0.959i)5-s + (−0.134 − 0.990i)8-s + (0.280 + 0.959i)10-s + (−0.163 + 0.986i)11-s + (−0.0448 − 0.998i)13-s + (−0.646 − 0.762i)16-s + (−0.575 + 0.817i)17-s + (0.913 + 0.406i)19-s + (0.753 + 0.657i)20-s + (0.393 + 0.919i)22-s + (0.946 + 0.323i)23-s + (−0.842 − 0.538i)25-s + (−0.575 − 0.817i)26-s + ⋯
L(s)  = 1  + (0.842 − 0.538i)2-s + (0.420 − 0.907i)4-s + (−0.280 + 0.959i)5-s + (−0.134 − 0.990i)8-s + (0.280 + 0.959i)10-s + (−0.163 + 0.986i)11-s + (−0.0448 − 0.998i)13-s + (−0.646 − 0.762i)16-s + (−0.575 + 0.817i)17-s + (0.913 + 0.406i)19-s + (0.753 + 0.657i)20-s + (0.393 + 0.919i)22-s + (0.946 + 0.323i)23-s + (−0.842 − 0.538i)25-s + (−0.575 − 0.817i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (845, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.971 - 0.237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.770069733 - 0.3331448669i\)
\(L(\frac12)\) \(\approx\) \(2.770069733 - 0.3331448669i\)
\(L(1)\) \(\approx\) \(1.593067182 - 0.2912903461i\)
\(L(1)\) \(\approx\) \(1.593067182 - 0.2912903461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.842 - 0.538i)T \)
5 \( 1 + (-0.280 + 0.959i)T \)
11 \( 1 + (-0.163 + 0.986i)T \)
13 \( 1 + (-0.0448 - 0.998i)T \)
17 \( 1 + (-0.575 + 0.817i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.946 + 0.323i)T \)
29 \( 1 + (0.858 - 0.512i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.873 - 0.486i)T \)
43 \( 1 + (0.473 + 0.880i)T \)
47 \( 1 + (0.842 - 0.538i)T \)
53 \( 1 + (0.420 - 0.907i)T \)
59 \( 1 + (-0.999 - 0.0299i)T \)
61 \( 1 + (-0.193 + 0.981i)T \)
67 \( 1 + (0.978 + 0.207i)T \)
71 \( 1 + (0.858 + 0.512i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.887 + 0.460i)T \)
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

−17.43130247784467622390599266631, −16.85149765076838833052519098427, −16.303414221940946389740601368824, −15.76322374570347256687796938166, −15.367973059079499910501818764110, −14.06018611039712590280831636472, −13.98825511382905580566577507427, −13.29922439769089324089403184770, −12.42895476297694143052134919047, −12.0652558021640323905292640003, −11.2524849342471305672042099328, −10.82273492640359155202319948911, −9.425251168110314248355831261453, −8.862897950046440380967512135119, −8.46469607620796240844075941231, −7.44859186931092745395177802815, −6.98017880983553291524380037250, −6.19280233734747904445269949379, −5.24873108462383011667989372545, −4.95544183836216028561979418319, −4.20733827808258792135176204809, −3.37544293335762176829996994832, −2.74299126437555469073934262542, −1.687608964849014981807380986729, −0.66521787647497527445111031654, 0.78293188045766862039047357125, 1.88265045388735069465374887967, 2.50130534268508955485147656729, 3.25815835552748428063630616643, 3.84244139141654026789201635547, 4.62152925688730954372711605226, 5.439912170067242899785019146231, 6.03650943318302993401744261775, 6.90367600417976503641002870423, 7.380474444361042448626394674131, 8.15356595189050954055918494342, 9.3226119761923871956368150550, 9.98375232668982899805742658236, 10.55749413693842045768144625111, 11.047259949567566580794368922489, 11.81822557992775897103073140259, 12.43563844848786813608431725036, 13.04707399647395175219951798528, 13.715374128411965257619746707, 14.45244421250607968460067245894, 15.08590776124081271920449010278, 15.38217855145986822681002744996, 16.021372321386789990738804193757, 17.15558670573598433770700516192, 17.88482236885566031181800473676

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