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Properties

Label 1-4729-4729.227-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.981 - 0.191i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.522 + 0.852i)2-s + (−0.999 + 0.0318i)3-s + (−0.453 − 0.891i)4-s + (0.830 + 0.556i)5-s + (0.495 − 0.868i)6-s + (0.999 + 0.0159i)7-s + (0.996 + 0.0796i)8-s + (0.997 − 0.0637i)9-s + (−0.908 + 0.417i)10-s + (0.996 + 0.0875i)11-s + (0.481 + 0.876i)12-s + (0.839 − 0.543i)13-s + (−0.536 + 0.843i)14-s + (−0.848 − 0.529i)15-s + (−0.589 + 0.808i)16-s + (−0.726 − 0.687i)17-s + ⋯
L(s)  = 1  + (−0.522 + 0.852i)2-s + (−0.999 + 0.0318i)3-s + (−0.453 − 0.891i)4-s + (0.830 + 0.556i)5-s + (0.495 − 0.868i)6-s + (0.999 + 0.0159i)7-s + (0.996 + 0.0796i)8-s + (0.997 − 0.0637i)9-s + (−0.908 + 0.417i)10-s + (0.996 + 0.0875i)11-s + (0.481 + 0.876i)12-s + (0.839 − 0.543i)13-s + (−0.536 + 0.843i)14-s + (−0.848 − 0.529i)15-s + (−0.589 + 0.808i)16-s + (−0.726 − 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.981 - 0.191i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.981 - 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297441203 - 0.1253273283i\)
\(L(\frac12)\) \(\approx\) \(1.297441203 - 0.1253273283i\)
\(L(1)\) \(\approx\) \(0.8266420280 + 0.2088557155i\)
\(L(1)\) \(\approx\) \(0.8266420280 + 0.2088557155i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.522 + 0.852i)T \)
3 \( 1 + (-0.999 + 0.0318i)T \)
5 \( 1 + (0.830 + 0.556i)T \)
7 \( 1 + (0.999 + 0.0159i)T \)
11 \( 1 + (0.996 + 0.0875i)T \)
13 \( 1 + (0.839 - 0.543i)T \)
17 \( 1 + (-0.726 - 0.687i)T \)
19 \( 1 + (0.721 - 0.692i)T \)
23 \( 1 + (0.582 - 0.812i)T \)
29 \( 1 + (-0.852 + 0.522i)T \)
31 \( 1 + (-0.127 - 0.991i)T \)
37 \( 1 + (-0.639 - 0.768i)T \)
41 \( 1 + (-0.798 + 0.601i)T \)
43 \( 1 + (-0.941 - 0.336i)T \)
47 \( 1 + (-0.952 - 0.305i)T \)
53 \( 1 + (0.988 - 0.150i)T \)
59 \( 1 + (-0.959 - 0.283i)T \)
61 \( 1 + (-0.474 - 0.880i)T \)
67 \( 1 + (0.981 - 0.190i)T \)
71 \( 1 + (-0.260 - 0.965i)T \)
73 \( 1 + (-0.576 + 0.817i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.975 - 0.221i)T \)
89 \( 1 + (0.343 + 0.939i)T \)
97 \( 1 + (-0.999 - 0.0159i)T \)
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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

−18.16376692000884747432612211413, −17.4703965476424368147888939044, −17.07313451616562549298655652367, −16.60772086853418976404447572734, −15.79456297899583150978114352980, −14.760521394789082379153589151624, −13.74331933422170509183940759566, −13.45716949605677981753118221159, −12.60153146095706030648293550356, −11.74608261736807252251888827273, −11.59547491327633500273431515810, −10.74516870596481271017512872416, −10.18832032854001329307500579310, −9.34717322906448874208095519438, −8.81766087464080085275193552658, −8.12422800179530584558646656941, −7.119866780342395630490866067516, −6.41418430133249549636172242217, −5.52854184302331153808020008657, −4.89033521381581010697921882918, −4.14234834064564074707248666490, −3.47921747194079707610705227011, −1.98081763236127311602517900121, −1.43432524944966081362416366705, −1.18431576837002347875639738548, 0.52946997334940759512578995650, 1.44661561116746624601931290122, 2.01166902473648145191407169112, 3.46406714952924960790448248293, 4.542023804576196187117597137, 5.12662108890531108338507787910, 5.695911300366234986584771958903, 6.51318203175070898578977705034, 6.90753556459528336863940365975, 7.60905055194743379604346446181, 8.65357679816812049652918420639, 9.25169030684625465802809263389, 9.913769052377925899119156901262, 10.8156783191330039895671870212, 11.09754514321335715126868209153, 11.761067910896692360303843284645, 12.99615053270067527520103122333, 13.56996823459652447016930105618, 14.203569276791842969038849564, 15.08707187002994848817440663138, 15.3230951959475006386764933645, 16.441533190089963892101438336075, 16.8434266372277672394069562090, 17.54495337846929550698200029894, 18.06582351669747503977769880858

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