↑

Properties

Label 1-229-229.197-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

Related objects

Downloads

Learn more

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: $\chi_{229} (197, \cdot )$
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 \( 1 + (0.0825 + 0.996i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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