Properties

Degree $1$
Conductor $283$
Sign $-0.944 - 0.327i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.296 − 0.955i)2-s + (−0.970 − 0.242i)3-s + (−0.824 + 0.565i)4-s + (−0.848 − 0.528i)5-s + (0.0556 + 0.998i)6-s + (0.556 − 0.830i)7-s + (0.784 + 0.619i)8-s + (0.882 + 0.470i)9-s + (−0.253 + 0.967i)10-s + (0.902 − 0.431i)11-s + (0.937 − 0.348i)12-s + (0.836 − 0.547i)13-s + (−0.958 − 0.285i)14-s + (0.695 + 0.718i)15-s + (0.359 − 0.933i)16-s + (0.726 + 0.687i)17-s + ⋯
L(s,χ)  = 1  + (−0.296 − 0.955i)2-s + (−0.970 − 0.242i)3-s + (−0.824 + 0.565i)4-s + (−0.848 − 0.528i)5-s + (0.0556 + 0.998i)6-s + (0.556 − 0.830i)7-s + (0.784 + 0.619i)8-s + (0.882 + 0.470i)9-s + (−0.253 + 0.967i)10-s + (0.902 − 0.431i)11-s + (0.937 − 0.348i)12-s + (0.836 − 0.547i)13-s + (−0.958 − 0.285i)14-s + (0.695 + 0.718i)15-s + (0.359 − 0.933i)16-s + (0.726 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.944 - 0.327i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.944 - 0.327i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.944 - 0.327i$
Motivic weight: \(0\)
Character: $\chi_{283} (11, \cdot )$
Sato-Tate group: $\mu(141)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.944 - 0.327i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.1058336547 - 0.6285532209i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.1058336547 - 0.6285532209i\)
\(L(\chi,1)\) \(\approx\) \(0.4556459910 - 0.4525776077i\)
\(L(1,\chi)\) \(\approx\) \(0.4556459910 - 0.4525776077i\)

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

−25.92392699902207457171075097198, −25.09059265617391898749621133228, −23.99671422838706892929421196839, −23.37909060008806404660349772482, −22.60280837762180823158041702359, −21.88258398936465201992283066255, −20.6580580878748612742681849086, −18.91260001032013085997932631444, −18.7192852564442421433392609661, −17.63108104531184337135284011452, −16.73468907068817994536721179986, −15.92949455221780794232700953433, −15.02302669997027626026782957167, −14.49572697818176037742454373619, −12.87756550652906285645005940349, −11.68797612441390743792924774099, −11.11855618213907565432843583430, −9.78139830830146177823256679773, −8.81423076761127770828697094408, −7.62471506118541304355626643478, −6.689925178579720347760283293463, −5.82298554294183928397159379918, −4.70005922282967471367175446149, −3.770485062309491538400000506725, −1.39336709555527073917338749741, 0.66270871886063125246987387855, 1.5250142161045562306592696062, 3.66094675391287551976015006773, 4.30774069551622617315713043528, 5.49118423622997710897271847239, 7.05951977141258245513241046959, 8.085887939497247994455804817433, 8.994098874070661405244894186932, 10.67731752454825913999441545367, 10.90695070510740125613024121088, 11.95475656075724733915768044981, 12.72107659110962412087684922371, 13.60358541100564185531174170686, 14.99787704464042075467203904188, 16.58897037388491289000151714334, 16.8877907044155553234939605412, 17.8751784385128400004830239101, 18.945077134799403001819820734659, 19.62487832471181311135745594160, 20.64075416714164326462421989982, 21.41496321614818353185696517847, 22.54629408441664036719751469513, 23.32325345543180084197985285331, 23.856101896878732627197794917703, 25.08886228963480089600740984489

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