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Properties

Label 1-4729-4729.227-r0-0-0
Degree 11
Conductor 47294729
Sign 0.981−0.191i0.981 - 0.191i
Analytic cond. 21.961321.9613
Root an. cond. 21.961321.9613
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(4729s/2 ΓR(s) L(s)=((0.981−0.191i) Λ‾(1−s)\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}
Λ(s)=(4729s/2 ΓR(s) L(s)=((0.981−0.191i) Λ‾(1−s)\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: 11
Conductor: 47294729
Sign: 0.981−0.191i0.981 - 0.191i
Analytic conductor: 21.961321.9613
Root analytic conductor: 21.961321.9613
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4729(227,⋅)\chi_{4729} (227, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 4729, (0: ), 0.981−0.191i)(1,\ 4729,\ (0:\ ),\ 0.981 - 0.191i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.297441203−0.1253273283i1.297441203 - 0.1253273283i
L(12)L(\frac12) ≈\approx 1.297441203−0.1253273283i1.297441203 - 0.1253273283i
L(1)L(1) ≈\approx 0.8266420280+0.2088557155i0.8266420280 + 0.2088557155i
L(1)L(1) ≈\approx 0.8266420280+0.2088557155i0.8266420280 + 0.2088557155i

Euler product

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