Properties

Degree $1$
Conductor $283$
Sign $0.836 - 0.547i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.593 + 0.805i)2-s + (−0.920 + 0.390i)3-s + (−0.296 − 0.955i)4-s + (−0.695 − 0.718i)5-s + (0.231 − 0.972i)6-s + (−0.0334 + 0.999i)7-s + (0.944 + 0.328i)8-s + (0.695 − 0.718i)9-s + (0.991 − 0.133i)10-s + (−0.296 + 0.955i)11-s + (0.645 + 0.763i)12-s + (0.231 − 0.972i)13-s + (−0.784 − 0.619i)14-s + (0.920 + 0.390i)15-s + (−0.824 + 0.565i)16-s + (−0.784 + 0.619i)17-s + ⋯
L(s,χ)  = 1  + (−0.593 + 0.805i)2-s + (−0.920 + 0.390i)3-s + (−0.296 − 0.955i)4-s + (−0.695 − 0.718i)5-s + (0.231 − 0.972i)6-s + (−0.0334 + 0.999i)7-s + (0.944 + 0.328i)8-s + (0.695 − 0.718i)9-s + (0.991 − 0.133i)10-s + (−0.296 + 0.955i)11-s + (0.645 + 0.763i)12-s + (0.231 − 0.972i)13-s + (−0.784 − 0.619i)14-s + (0.920 + 0.390i)15-s + (−0.824 + 0.565i)16-s + (−0.784 + 0.619i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.836 - 0.547i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.836 - 0.547i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.836 - 0.547i$
Motivic weight: \(0\)
Character: $\chi_{283} (142, \cdot )$
Sato-Tate group: $\mu(94)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.836 - 0.547i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.2189603924 - 0.06526674253i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.2189603924 - 0.06526674253i\)
\(L(\chi,1)\) \(\approx\) \(0.3893900587 + 0.1889998116i\)
\(L(1,\chi)\) \(\approx\) \(0.3893900587 + 0.1889998116i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.899977782853536494492541424341, −24.144501201286426109946034716668, −23.63889591464298413559882866113, −22.51872982580268858264721135535, −21.982024610569102846416179306940, −20.86752973792134171048139697533, −19.661020014909941262515904828182, −19.058906245827872761598824590167, −18.25803609787001600902475823990, −17.37227776550688230831440979168, −16.47778024124490318691892057182, −15.70832931794450168546328072907, −13.778237801811293291544189074601, −13.333030178679834610712992701238, −11.84334657794374986314181857754, −11.25215597406056128352353006933, −10.753932136373422452673471618302, −9.584257488489985233550019976976, −8.1173347449521516329022937270, −7.253750580997286427098630848567, −6.43763318303153017676050560774, −4.58544040857127066820621985612, −3.7078691689215316953711479322, −2.26735783013002773480286135884, −0.72484908733137925235845147757, 0.15341873598968580007453650323, 1.70458310917434378794804008669, 4.01628526700549473131109659862, 5.10368265240326400876779457824, 5.75656515887241215430163923007, 6.94526700465288230964193023533, 8.157287057835869843232269208800, 8.95061608499577187691066406289, 10.09446379319469585856926476191, 10.93764018636866356275407346465, 12.291617558961890727085492881721, 12.78921930039494707154266316296, 14.73808041375679603279408285907, 15.460153900304690135720424377986, 16.013447138746409161502363508596, 16.91638859768186616615564404741, 17.92379587150536370377245941171, 18.43555680429239208663374246881, 19.762849264011805234116381714244, 20.58871523883249353837919397618, 21.91835267801844869568553735909, 22.867317597730680803036274402659, 23.46904598594310496530290231113, 24.434699890579526252519140752558, 25.142059216232301931450478086384

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