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Properties

Label 1-283-283.194-r1-0-0
Degree $1$
Conductor $283$
Sign $0.0408 + 0.999i$
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  + (0.979 + 0.199i)2-s + (−0.0111 + 0.999i)3-s + (0.920 + 0.390i)4-s + (−0.519 − 0.854i)5-s + (−0.210 + 0.977i)6-s + (0.999 − 0.0445i)7-s + (0.824 + 0.565i)8-s + (−0.999 − 0.0222i)9-s + (−0.338 − 0.940i)10-s + (−0.122 + 0.992i)11-s + (−0.400 + 0.916i)12-s + (0.951 − 0.306i)13-s + (0.987 + 0.155i)14-s + (0.860 − 0.509i)15-s + (0.695 + 0.718i)16-s + (−0.628 − 0.777i)17-s + ⋯
L(s)  = 1  + (0.979 + 0.199i)2-s + (−0.0111 + 0.999i)3-s + (0.920 + 0.390i)4-s + (−0.519 − 0.854i)5-s + (−0.210 + 0.977i)6-s + (0.999 − 0.0445i)7-s + (0.824 + 0.565i)8-s + (−0.999 − 0.0222i)9-s + (−0.338 − 0.940i)10-s + (−0.122 + 0.992i)11-s + (−0.400 + 0.916i)12-s + (0.951 − 0.306i)13-s + (0.987 + 0.155i)14-s + (0.860 − 0.509i)15-s + (0.695 + 0.718i)16-s + (−0.628 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0408 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0408 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.0408 + 0.999i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.0408 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.769625329 + 2.658725822i\)
\(L(\frac12)\) \(\approx\) \(2.769625329 + 2.658725822i\)
\(L(1)\) \(\approx\) \(1.864152992 + 0.8845261308i\)
\(L(1)\) \(\approx\) \(1.864152992 + 0.8845261308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.979 + 0.199i)T \)
3 \( 1 + (-0.0111 + 0.999i)T \)
5 \( 1 + (-0.519 - 0.854i)T \)
7 \( 1 + (0.999 - 0.0445i)T \)
11 \( 1 + (-0.122 + 0.992i)T \)
13 \( 1 + (0.951 - 0.306i)T \)
17 \( 1 + (-0.628 - 0.777i)T \)
19 \( 1 + (0.741 + 0.670i)T \)
23 \( 1 + (0.441 + 0.897i)T \)
29 \( 1 + (-0.296 + 0.955i)T \)
31 \( 1 + (0.929 + 0.369i)T \)
37 \( 1 + (0.610 - 0.791i)T \)
41 \( 1 + (0.902 + 0.431i)T \)
43 \( 1 + (-0.964 - 0.264i)T \)
47 \( 1 + (0.210 + 0.977i)T \)
53 \( 1 + (-0.695 + 0.718i)T \)
59 \( 1 + (-0.575 - 0.818i)T \)
61 \( 1 + (-0.645 - 0.763i)T \)
67 \( 1 + (-0.593 + 0.805i)T \)
71 \( 1 + (0.920 - 0.390i)T \)
73 \( 1 + (-0.929 + 0.369i)T \)
79 \( 1 + (-0.964 + 0.264i)T \)
83 \( 1 + (0.882 - 0.470i)T \)
89 \( 1 + (-0.711 + 0.703i)T \)
97 \( 1 + (0.556 - 0.830i)T \)
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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

−24.60784795326696716429829294099, −24.17863812157396059436290758253, −23.39441525072983012101373533248, −22.59428200549920095294304262315, −21.63969306853881722645190063890, −20.6713334088671008554592487764, −19.64442679378264191922314472915, −18.83804278690482489135197915326, −18.12267789716323572225376448176, −16.833646413775696560426143617870, −15.547770421366471096136234693980, −14.736483121792107849608050611463, −13.80646087938417008161488744080, −13.275654473418157010083966355928, −11.83441850180718226268142106757, −11.33743613752504794083620544540, −10.674813285029522063588368346977, −8.530721049574891246936545699875, −7.69337947803337471760462121384, −6.58087234925165447111536903404, −5.90690579022073715957048847994, −4.46555246409204753270138052003, −3.22225637069164582270291638180, −2.23323559171437152144088115136, −0.913391974525250675562297981634, 1.47263312117357575251585482597, 3.14299385242448693541417009562, 4.28492443366299107467372796666, 4.86959505440246565516550303790, 5.71258045193436201064068304999, 7.407667146790902866620063694501, 8.28098942896346046620040719746, 9.42731308545054988184964997762, 10.88227003047883059479222189665, 11.51523986334943308792599499198, 12.46447693348036356980512489899, 13.64348452058675070999584300848, 14.56988467813496486227682696261, 15.54898116566920032952043224422, 15.95518761062819649470361246664, 17.02999956064573847257492474858, 17.93911409178016003450966191127, 19.865780281089841573136770528291, 20.53446040439604353340983053602, 20.900157315774985572242226015573, 21.92937395283812238932912202158, 23.104542835513630530894435047625, 23.39071230612727700764532958879, 24.7116668594379403259186394759, 25.26573618554630050153849025575

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