Properties

Degree $1$
Conductor $283$
Sign $0.728 - 0.684i$
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 + (0.122 + 0.992i)3-s + (−0.296 + 0.955i)4-s + (−0.274 − 0.961i)5-s + (0.726 − 0.687i)6-s + (0.882 + 0.470i)7-s + (0.944 − 0.328i)8-s + (−0.970 + 0.242i)9-s + (−0.610 + 0.791i)10-s + (0.975 + 0.220i)11-s + (−0.984 − 0.177i)12-s + (−0.958 − 0.285i)13-s + (−0.144 − 0.989i)14-s + (0.920 − 0.390i)15-s + (−0.824 − 0.565i)16-s + (0.929 − 0.369i)17-s + ⋯
L(s,χ)  = 1  + (−0.593 − 0.805i)2-s + (0.122 + 0.992i)3-s + (−0.296 + 0.955i)4-s + (−0.274 − 0.961i)5-s + (0.726 − 0.687i)6-s + (0.882 + 0.470i)7-s + (0.944 − 0.328i)8-s + (−0.970 + 0.242i)9-s + (−0.610 + 0.791i)10-s + (0.975 + 0.220i)11-s + (−0.984 − 0.177i)12-s + (−0.958 − 0.285i)13-s + (−0.144 − 0.989i)14-s + (0.920 − 0.390i)15-s + (−0.824 − 0.565i)16-s + (0.929 − 0.369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.728 - 0.684i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.728 - 0.684i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.728 - 0.684i$
Motivic weight: \(0\)
Character: $\chi_{283} (88, \cdot )$
Sato-Tate group: $\mu(282)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.319249451 - 0.5224762158i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.319249451 - 0.5224762158i\)
\(L(\chi,1)\) \(\approx\) \(0.8811783653 - 0.1591842805i\)
\(L(1,\chi)\) \(\approx\) \(0.8811783653 - 0.1591842805i\)

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

−25.465929374491269245761458994224, −24.519918635040454484261132100609, −23.92047177943822613902860407120, −23.044664650691526236080251015765, −22.267559611971258707436190993478, −20.665010515518427987962455847367, −19.44502731716665203001591610859, −19.03094071278915129644417667201, −18.15555438264320100569661795194, −17.21185501824884227818637520861, −16.69171564715363194423911675199, −14.85653655409635094339869163898, −14.54997395753040044392080887727, −13.87258896676099126684861179161, −12.29920393448294860449767125517, −11.27526408061784034883236935276, −10.35787244939261184518011069295, −9.02244718073248895903042431003, −7.92346855313291657779285434427, −7.29584301571307389371811695054, −6.53234842017792956716749400847, −5.39261268010954315838910185301, −3.79766438828370455681228294180, −2.07748936154506284937353018570, −0.96533409264420539076082939941, 0.69025018350942051420964644907, 2.12799632905948281760020853654, 3.47255176463280959054095389031, 4.61424501149874945764856004645, 5.22417018641574846421727538025, 7.43166606899223532208174351346, 8.49842220402761929995856371452, 9.1726960291895979907745054249, 9.921868366915599260166688422241, 11.26860558984534220237772207788, 11.801032275421383188120402320679, 12.78391709259879629123878177942, 14.19836808659838859091045778706, 15.15233937650672014876235987892, 16.27185169231055277316566677623, 17.15909875991490801427879179607, 17.64128290544777280800509429308, 19.285167924241561594365413832579, 19.82245818949700822175407884844, 20.80030413581478736507945753851, 21.31736443670071650610809202007, 22.15900565054173539909703544392, 23.196301175679208486220535994189, 24.71153652081507131424122021717, 25.24135689602470832709440559371

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