Properties

Degree 1
Conductor 229
Sign $-0.575 + 0.817i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $-0.575 + 0.817i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (93, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ -0.575 + 0.817i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2574754383 + 0.4958333016i$
$L(\frac12,\chi)$  $\approx$  $0.2574754383 + 0.4958333016i$
$L(\chi,1)$  $\approx$  1.042639312 - 0.1283467914i
$L(1,\chi)$  $\approx$  1.042639312 - 0.1283467914i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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