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Properties

Label 1-229-229.227-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.996 - 0.0861i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (−0.915 + 0.401i)2-s + (0.546 + 0.837i)3-s + (0.677 − 0.735i)4-s + (0.986 + 0.164i)5-s + (−0.837 − 0.546i)6-s + (−0.614 + 0.789i)7-s + (−0.324 + 0.945i)8-s + (−0.401 + 0.915i)9-s + (−0.969 + 0.245i)10-s + (0.879 − 0.475i)11-s + (0.986 + 0.164i)12-s + (−0.164 + 0.986i)13-s + (0.245 − 0.969i)14-s + (0.401 + 0.915i)15-s + (−0.0825 − 0.996i)16-s + (−0.986 − 0.164i)17-s + ⋯
L(s)  = 1  + (−0.915 + 0.401i)2-s + (0.546 + 0.837i)3-s + (0.677 − 0.735i)4-s + (0.986 + 0.164i)5-s + (−0.837 − 0.546i)6-s + (−0.614 + 0.789i)7-s + (−0.324 + 0.945i)8-s + (−0.401 + 0.915i)9-s + (−0.969 + 0.245i)10-s + (0.879 − 0.475i)11-s + (0.986 + 0.164i)12-s + (−0.164 + 0.986i)13-s + (0.245 − 0.969i)14-s + (0.401 + 0.915i)15-s + (−0.0825 − 0.996i)16-s + (−0.986 − 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.996 - 0.0861i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.996 - 0.0861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04905557393 + 1.137062644i\)
\(L(\frac12)\) \(\approx\) \(-0.04905557393 + 1.137062644i\)
\(L(1)\) \(\approx\) \(0.6598720915 + 0.5597409888i\)
\(L(1)\) \(\approx\) \(0.6598720915 + 0.5597409888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.915 + 0.401i)T \)
3 \( 1 + (0.546 + 0.837i)T \)
5 \( 1 + (0.986 + 0.164i)T \)
7 \( 1 + (-0.614 + 0.789i)T \)
11 \( 1 + (0.879 - 0.475i)T \)
13 \( 1 + (-0.164 + 0.986i)T \)
17 \( 1 + (-0.986 - 0.164i)T \)
19 \( 1 + (-0.986 + 0.164i)T \)
23 \( 1 + (-0.969 - 0.245i)T \)
29 \( 1 + (0.614 - 0.789i)T \)
31 \( 1 + (0.475 + 0.879i)T \)
37 \( 1 + (-0.0825 + 0.996i)T \)
41 \( 1 + (0.915 - 0.401i)T \)
43 \( 1 + (-0.0825 + 0.996i)T \)
47 \( 1 + (-0.915 - 0.401i)T \)
53 \( 1 + (0.546 + 0.837i)T \)
59 \( 1 + (-0.996 + 0.0825i)T \)
61 \( 1 + (-0.401 + 0.915i)T \)
67 \( 1 + (-0.915 - 0.401i)T \)
71 \( 1 + (0.879 + 0.475i)T \)
73 \( 1 + (0.324 - 0.945i)T \)
79 \( 1 + (0.614 + 0.789i)T \)
83 \( 1 + (-0.0825 - 0.996i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.945 + 0.324i)T \)
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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

−25.76817953148810219532236146812, −25.0056935604634254377777543874, −24.24610850544467898727671621075, −22.79687567949567968531679480211, −21.75885358642811421483410386338, −20.533234113447210446025988095104, −19.91260057125772754215782121136, −19.30488926475791652388451079407, −17.86243143096065125636772621534, −17.60342321796952638635946164581, −16.622187531925435451367768431101, −15.19768789967504023222731713199, −13.937653058473785976160872421179, −12.97201757029250998116598634389, −12.39337248536618232454166020304, −10.8769933948890276996981306090, −9.85443827875458376775541353867, −9.09336674879372244384713247711, −8.03048205322681360573129170717, −6.854250760886880195305968017828, −6.237361368819210981298567913777, −3.96644341734941614589063583808, −2.628029655918101129394473275775, −1.65638416919573853723892636171, −0.43463270778145510773530029874, 1.86859123062587291525117954261, 2.780593222184314247845277602944, 4.541133364787728090645486661851, 6.03213778537063548279313948413, 6.5978117860025751380952118049, 8.42879362150924884974644393329, 9.09689635892592858037189982913, 9.74742544675966573008601723179, 10.709773053844803856503135713097, 11.88614561858664320719418415057, 13.64510578436267579440918530986, 14.42952775094207087304503840430, 15.35757553301193084614184579066, 16.32465412676701714453622596138, 17.01402213122328747257069024919, 18.133182938204450489996472493128, 19.217938905175056789773143927984, 19.77085313570928063411240725384, 21.14772374868589393059156077132, 21.70835725314238068604272215830, 22.70687520345663557753748193730, 24.44297202510128417354892290909, 24.974989894432369509273548046077, 25.90691130276667859916585549273, 26.38029051522484487444260982750

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