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Properties

Label 1-283-283.88-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

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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: $\chi_{283} (88, \cdot )$
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 \( 1 + (-0.210 - 0.977i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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