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Properties

Label 1-283-283.84-r1-0-0
Degree $1$
Conductor $283$
Sign $0.233 - 0.972i$
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  + (−0.784 − 0.619i)2-s + (0.741 + 0.670i)3-s + (0.231 + 0.972i)4-s + (−0.100 + 0.994i)5-s + (−0.166 − 0.986i)6-s + (−0.979 − 0.199i)7-s + (0.420 − 0.907i)8-s + (0.100 + 0.994i)9-s + (0.695 − 0.718i)10-s + (0.231 − 0.972i)11-s + (−0.480 + 0.876i)12-s + (−0.166 − 0.986i)13-s + (0.645 + 0.763i)14-s + (−0.741 + 0.670i)15-s + (−0.892 + 0.451i)16-s + (0.645 − 0.763i)17-s + ⋯
L(s)  = 1  + (−0.784 − 0.619i)2-s + (0.741 + 0.670i)3-s + (0.231 + 0.972i)4-s + (−0.100 + 0.994i)5-s + (−0.166 − 0.986i)6-s + (−0.979 − 0.199i)7-s + (0.420 − 0.907i)8-s + (0.100 + 0.994i)9-s + (0.695 − 0.718i)10-s + (0.231 − 0.972i)11-s + (−0.480 + 0.876i)12-s + (−0.166 − 0.986i)13-s + (0.645 + 0.763i)14-s + (−0.741 + 0.670i)15-s + (−0.892 + 0.451i)16-s + (0.645 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.233 - 0.972i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.233 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7464053660 - 0.5886376397i\)
\(L(\frac12)\) \(\approx\) \(0.7464053660 - 0.5886376397i\)
\(L(1)\) \(\approx\) \(0.7842324960 - 0.05105118255i\)
\(L(1)\) \(\approx\) \(0.7842324960 - 0.05105118255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.784 - 0.619i)T \)
3 \( 1 + (0.741 + 0.670i)T \)
5 \( 1 + (-0.100 + 0.994i)T \)
7 \( 1 + (-0.979 - 0.199i)T \)
11 \( 1 + (0.231 - 0.972i)T \)
13 \( 1 + (-0.166 - 0.986i)T \)
17 \( 1 + (0.645 - 0.763i)T \)
19 \( 1 + (0.166 + 0.986i)T \)
23 \( 1 + (-0.296 - 0.955i)T \)
29 \( 1 + (-0.538 - 0.842i)T \)
31 \( 1 + (-0.991 + 0.133i)T \)
37 \( 1 + (0.824 - 0.565i)T \)
41 \( 1 + (-0.420 - 0.907i)T \)
43 \( 1 + (-0.359 + 0.933i)T \)
47 \( 1 + (0.166 - 0.986i)T \)
53 \( 1 + (0.892 + 0.451i)T \)
59 \( 1 + (0.920 + 0.390i)T \)
61 \( 1 + (0.695 + 0.718i)T \)
67 \( 1 + (-0.480 - 0.876i)T \)
71 \( 1 + (0.231 - 0.972i)T \)
73 \( 1 + (0.991 + 0.133i)T \)
79 \( 1 + (-0.359 - 0.933i)T \)
83 \( 1 + (0.593 - 0.805i)T \)
89 \( 1 + (0.359 - 0.933i)T \)
97 \( 1 + (-0.296 - 0.955i)T \)
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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

−25.70702328571691406257636086426, −24.87106902478646974842761859600, −23.72781848365156638428758272837, −23.59911688623169640775220643686, −21.904154694943522451687414209, −20.51933355173223661750178977222, −19.80158305571839949003523343014, −19.27784897011709965570447886508, −18.28301311598556046160969439245, −17.258620885756388185995892226403, −16.45426722385274517878569720822, −15.46468807776071171326742716705, −14.634706658207444170576651210020, −13.45064295388775209604934922427, −12.627786825476580301638407415746, −11.61337570209754154742364924746, −9.71253714151046933207135380727, −9.37448956723421014273089465046, −8.434994963215824809182376887039, −7.35082701381541219017369479023, −6.61615065217805436240131641923, −5.36833826828318535482851210535, −3.88581941386141495409879453123, −2.15581922560505420928641278398, −1.16293002651433235602490573367, 0.36983048276287986678319109675, 2.3885475593240430905750715487, 3.27217494416547493717513806672, 3.81158630450577099995000472318, 5.87134897258740227417353541781, 7.27022580152962355159732190021, 8.07110748075989790687607253874, 9.22318059643301986782106884212, 10.13716867726810220268464242001, 10.59220171000674993314838134659, 11.7723755060522919305626241855, 13.05517901448330301231589772357, 14.00891916489794783297513897544, 15.0672270880166784157748990434, 16.204213740468532119570570086992, 16.66861170929405038550394070168, 18.26303534640453568523244245074, 18.88400660076740600845837988192, 19.67203221301359031855560857003, 20.43297032666750501089042332879, 21.42835049882035285945703672302, 22.33354107340217045251436949615, 22.80935433352556437862051550503, 24.81995289927868495136237141237, 25.467017088545491662595366536616

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