Properties

Degree 1
Conductor 283
Sign $0.233 - 0.972i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.784 − 0.619i)2-s + (0.741 + 0.670i)3-s + (0.231 + 0.972i)4-s + (−0.100 + 0.994i)5-s + (−0.166 − 0.986i)6-s + (−0.979 − 0.199i)7-s + (0.420 − 0.907i)8-s + (0.100 + 0.994i)9-s + (0.695 − 0.718i)10-s + (0.231 − 0.972i)11-s + (−0.480 + 0.876i)12-s + (−0.166 − 0.986i)13-s + (0.645 + 0.763i)14-s + (−0.741 + 0.670i)15-s + (−0.892 + 0.451i)16-s + (0.645 − 0.763i)17-s + ⋯
L(s,χ)  = 1  + (−0.784 − 0.619i)2-s + (0.741 + 0.670i)3-s + (0.231 + 0.972i)4-s + (−0.100 + 0.994i)5-s + (−0.166 − 0.986i)6-s + (−0.979 − 0.199i)7-s + (0.420 − 0.907i)8-s + (0.100 + 0.994i)9-s + (0.695 − 0.718i)10-s + (0.231 − 0.972i)11-s + (−0.480 + 0.876i)12-s + (−0.166 − 0.986i)13-s + (0.645 + 0.763i)14-s + (−0.741 + 0.670i)15-s + (−0.892 + 0.451i)16-s + (0.645 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.233 - 0.972i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.233 - 0.972i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $0.233 - 0.972i$
motivic weight  =  \(0\)
character  :  $\chi_{283} (84, \cdot )$
Sato-Tate  :  $\mu(94)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 283,\ (1:\ ),\ 0.233 - 0.972i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7464053660 - 0.5886376397i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7464053660 - 0.5886376397i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7842324960 - 0.05105118255i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7842324960 - 0.05105118255i\)

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

−25.70702328571691406257636086426, −24.87106902478646974842761859600, −23.72781848365156638428758272837, −23.59911688623169640775220643686, −21.904154694943522451687414209, −20.51933355173223661750178977222, −19.80158305571839949003523343014, −19.27784897011709965570447886508, −18.28301311598556046160969439245, −17.258620885756388185995892226403, −16.45426722385274517878569720822, −15.46468807776071171326742716705, −14.634706658207444170576651210020, −13.45064295388775209604934922427, −12.627786825476580301638407415746, −11.61337570209754154742364924746, −9.71253714151046933207135380727, −9.37448956723421014273089465046, −8.434994963215824809182376887039, −7.35082701381541219017369479023, −6.61615065217805436240131641923, −5.36833826828318535482851210535, −3.88581941386141495409879453123, −2.15581922560505420928641278398, −1.16293002651433235602490573367, 0.36983048276287986678319109675, 2.3885475593240430905750715487, 3.27217494416547493717513806672, 3.81158630450577099995000472318, 5.87134897258740227417353541781, 7.27022580152962355159732190021, 8.07110748075989790687607253874, 9.22318059643301986782106884212, 10.13716867726810220268464242001, 10.59220171000674993314838134659, 11.7723755060522919305626241855, 13.05517901448330301231589772357, 14.00891916489794783297513897544, 15.0672270880166784157748990434, 16.204213740468532119570570086992, 16.66861170929405038550394070168, 18.26303534640453568523244245074, 18.88400660076740600845837988192, 19.67203221301359031855560857003, 20.43297032666750501089042332879, 21.42835049882035285945703672302, 22.33354107340217045251436949615, 22.80935433352556437862051550503, 24.81995289927868495136237141237, 25.467017088545491662595366536616

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