↑

Properties

Label 1-283-283.119-r1-0-0
Degree $1$
Conductor $283$
Sign $0.728 + 0.684i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.728 + 0.684i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.593 + 0.805i)T \)
3 \( 1 + (0.122 - 0.992i)T \)
5 \( 1 + (-0.274 + 0.961i)T \)
7 \( 1 + (0.882 - 0.470i)T \)
11 \( 1 + (0.975 - 0.220i)T \)
13 \( 1 + (-0.958 + 0.285i)T \)
17 \( 1 + (0.929 + 0.369i)T \)
19 \( 1 + (-0.231 + 0.972i)T \)
23 \( 1 + (0.951 - 0.306i)T \)
29 \( 1 + (-0.166 + 0.986i)T \)
31 \( 1 + (-0.519 - 0.854i)T \)
37 \( 1 + (-0.811 + 0.584i)T \)
41 \( 1 + (0.188 - 0.982i)T \)
43 \( 1 + (0.979 - 0.199i)T \)
47 \( 1 + (-0.726 + 0.687i)T \)
53 \( 1 + (0.824 + 0.565i)T \)
59 \( 1 + (0.441 + 0.897i)T \)
61 \( 1 + (0.991 + 0.133i)T \)
67 \( 1 + (0.645 - 0.763i)T \)
71 \( 1 + (-0.296 + 0.955i)T \)
73 \( 1 + (0.519 - 0.854i)T \)
79 \( 1 + (0.979 + 0.199i)T \)
83 \( 1 + (0.628 + 0.777i)T \)
89 \( 1 + (0.662 + 0.749i)T \)
97 \( 1 + (-0.210 + 0.977i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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