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Properties

Label 1-229-229.93-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.575 + 0.817i$
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  + (0.475 − 0.879i)2-s + (0.245 + 0.969i)3-s + (−0.546 − 0.837i)4-s + (0.677 − 0.735i)5-s + (0.969 + 0.245i)6-s + (0.164 + 0.986i)7-s + (−0.996 + 0.0825i)8-s + (−0.879 + 0.475i)9-s + (−0.324 − 0.945i)10-s + (−0.789 + 0.614i)11-s + (0.677 − 0.735i)12-s + (−0.735 − 0.677i)13-s + (0.945 + 0.324i)14-s + (0.879 + 0.475i)15-s + (−0.401 + 0.915i)16-s + (−0.677 + 0.735i)17-s + ⋯
L(s)  = 1  + (0.475 − 0.879i)2-s + (0.245 + 0.969i)3-s + (−0.546 − 0.837i)4-s + (0.677 − 0.735i)5-s + (0.969 + 0.245i)6-s + (0.164 + 0.986i)7-s + (−0.996 + 0.0825i)8-s + (−0.879 + 0.475i)9-s + (−0.324 − 0.945i)10-s + (−0.789 + 0.614i)11-s + (0.677 − 0.735i)12-s + (−0.735 − 0.677i)13-s + (0.945 + 0.324i)14-s + (0.879 + 0.475i)15-s + (−0.401 + 0.915i)16-s + (−0.677 + 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.575 + 0.817i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.575 + 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2574754383 + 0.4958333016i\)
\(L(\frac12)\) \(\approx\) \(0.2574754383 + 0.4958333016i\)
\(L(1)\) \(\approx\) \(1.042639312 - 0.1283467914i\)
\(L(1)\) \(\approx\) \(1.042639312 - 0.1283467914i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.475 - 0.879i)T \)
3 \( 1 + (0.245 + 0.969i)T \)
5 \( 1 + (0.677 - 0.735i)T \)
7 \( 1 + (0.164 + 0.986i)T \)
11 \( 1 + (-0.789 + 0.614i)T \)
13 \( 1 + (-0.735 - 0.677i)T \)
17 \( 1 + (-0.677 + 0.735i)T \)
19 \( 1 + (-0.677 - 0.735i)T \)
23 \( 1 + (-0.324 + 0.945i)T \)
29 \( 1 + (-0.164 - 0.986i)T \)
31 \( 1 + (0.614 + 0.789i)T \)
37 \( 1 + (-0.401 - 0.915i)T \)
41 \( 1 + (-0.475 + 0.879i)T \)
43 \( 1 + (-0.401 - 0.915i)T \)
47 \( 1 + (0.475 + 0.879i)T \)
53 \( 1 + (0.245 + 0.969i)T \)
59 \( 1 + (-0.915 - 0.401i)T \)
61 \( 1 + (-0.879 + 0.475i)T \)
67 \( 1 + (0.475 + 0.879i)T \)
71 \( 1 + (-0.789 - 0.614i)T \)
73 \( 1 + (0.996 - 0.0825i)T \)
79 \( 1 + (-0.164 + 0.986i)T \)
83 \( 1 + (-0.401 + 0.915i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.0825 - 0.996i)T \)
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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

−25.88861148822520114786379627683, −24.66822314002580067701658327063, −24.12502367102572216979901071983, −23.199204886981149778352751536606, −22.419328846105128424573106192435, −21.305511128650819055682568263933, −20.32493205823712734151318383179, −18.84862174153148014599539046009, −18.25321175552369083921009083754, −17.23691443492371749256968768133, −16.55894029181081792666744950872, −15.02251195535937555542093174577, −14.110328743712931850852453188185, −13.685890524405378419059584565, −12.77335059526962216333449619241, −11.49794203413835341052166977745, −10.1959677918246133440132658001, −8.773793495251803062202901986615, −7.696238412430629032566930843980, −6.88223476621038326614388349674, −6.178444617600778928134271525300, −4.82771894512242150756431922421, −3.31078352946468591892309022210, −2.16459241809159261692190654842, −0.13342007833484271730469888503, 2.00946841384046959197846204919, 2.76575791848117017703012026203, 4.38006446720050391487344439082, 5.132158005214836933013094259941, 5.907837339938678651998661479119, 8.28031910701583691827480521323, 9.18232258574059909903215256214, 9.95813386967823867737078854427, 10.838296482902706134728724746963, 12.12215763006204189550499724714, 12.939657276211717682681215431195, 13.91890037998127634681310388632, 15.26937799060095310920171760990, 15.45962081143833950643427620243, 17.25637078249245456898895414097, 17.91171032851237461767975634933, 19.40907233342257820271214565319, 20.126229479263927990529944027, 21.10975523249181924409330010164, 21.592114013262849240337870498023, 22.299713329877406042051434570936, 23.50035745187662071282700695954, 24.6221661097579741295692890749, 25.49943639360260230632002655773, 26.62502322383412079565549419344

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