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Properties

Label 1-283-283.73-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.964 - 0.262i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (0.480 − 0.876i)2-s + (0.726 − 0.687i)3-s + (−0.538 − 0.842i)4-s + (0.836 − 0.547i)5-s + (−0.253 − 0.967i)6-s + (−0.993 + 0.111i)7-s + (−0.997 + 0.0667i)8-s + (0.0556 − 0.998i)9-s + (−0.0779 − 0.996i)10-s + (−0.460 − 0.887i)11-s + (−0.970 − 0.242i)12-s + (−0.711 + 0.703i)13-s + (−0.380 + 0.924i)14-s + (0.231 − 0.972i)15-s + (−0.420 + 0.907i)16-s + (−0.610 + 0.791i)17-s + ⋯
L(s)  = 1  + (0.480 − 0.876i)2-s + (0.726 − 0.687i)3-s + (−0.538 − 0.842i)4-s + (0.836 − 0.547i)5-s + (−0.253 − 0.967i)6-s + (−0.993 + 0.111i)7-s + (−0.997 + 0.0667i)8-s + (0.0556 − 0.998i)9-s + (−0.0779 − 0.996i)10-s + (−0.460 − 0.887i)11-s + (−0.970 − 0.242i)12-s + (−0.711 + 0.703i)13-s + (−0.380 + 0.924i)14-s + (0.231 − 0.972i)15-s + (−0.420 + 0.907i)16-s + (−0.610 + 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.964 - 0.262i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.964 - 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2318788439 - 1.734127479i\)
\(L(\frac12)\) \(\approx\) \(0.2318788439 - 1.734127479i\)
\(L(1)\) \(\approx\) \(0.9163442352 - 1.185638031i\)
\(L(1)\) \(\approx\) \(0.9163442352 - 1.185638031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.480 - 0.876i)T \)
3 \( 1 + (0.726 - 0.687i)T \)
5 \( 1 + (0.836 - 0.547i)T \)
7 \( 1 + (-0.993 + 0.111i)T \)
11 \( 1 + (-0.460 - 0.887i)T \)
13 \( 1 + (-0.711 + 0.703i)T \)
17 \( 1 + (-0.610 + 0.791i)T \)
19 \( 1 + (0.964 + 0.264i)T \)
23 \( 1 + (0.937 - 0.348i)T \)
29 \( 1 + (-0.0334 - 0.999i)T \)
31 \( 1 + (0.811 - 0.584i)T \)
37 \( 1 + (0.756 - 0.654i)T \)
41 \( 1 + (0.441 + 0.897i)T \)
43 \( 1 + (0.784 + 0.619i)T \)
47 \( 1 + (-0.253 + 0.967i)T \)
53 \( 1 + (-0.420 - 0.907i)T \)
59 \( 1 + (-0.679 - 0.734i)T \)
61 \( 1 + (-0.824 - 0.565i)T \)
67 \( 1 + (0.695 + 0.718i)T \)
71 \( 1 + (-0.538 + 0.842i)T \)
73 \( 1 + (0.811 + 0.584i)T \)
79 \( 1 + (0.784 - 0.619i)T \)
83 \( 1 + (-0.338 + 0.940i)T \)
89 \( 1 + (-0.929 + 0.369i)T \)
97 \( 1 + (-0.770 - 0.637i)T \)
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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

−25.789005966844825928122620548637, −25.30324342057433906511538506211, −24.54417905257189419049968250377, −22.94870503696456670802144270796, −22.446481659622955546164358684192, −21.750709020243157771358534657674, −20.71279207941454179750929930104, −19.84200360256499509192215472031, −18.49287712402447916708181426826, −17.5954163181785033064397628243, −16.6132815525654472029320947685, −15.55190824584610555714666963762, −15.108303513889137569679059288383, −13.94464663456143760341125723351, −13.39735980065960756983859877635, −12.41425911674171389546277066621, −10.61269551542625053766820770623, −9.63668232797161683431658067007, −9.10106764737349171592534641941, −7.50899102255021102574391132444, −6.91840689155882960175371886551, −5.48932872145559925974392905451, −4.70469728112381555097982578384, −3.168391647613069718755168321084, −2.658464902882791238437198798373, 0.93543620146692128497170153979, 2.28547035545114547278144087549, 3.00665481384050438255742883137, 4.35063297071253933462545769231, 5.802569211091590982426940808210, 6.51713053313856184875239476250, 8.20662015936397796667074676572, 9.36508129543245020640390098561, 9.71866940886584576886658614427, 11.20957639423490252215313201335, 12.455891525581169939289519718970, 13.034884263200105454813740971254, 13.70318480868223295720722063877, 14.530074330222777742534929793175, 15.78574715916378731537702299740, 17.08727797365929062618102954802, 18.21251268749604815267143864448, 19.11948440846743936009693534732, 19.6138956276354690984644497052, 20.70064916916692667831250419795, 21.3791804927465160180355216951, 22.258022168419822033764207233249, 23.368916214779109797507440393218, 24.45473671703709853423100585835, 24.754232577074774444714343138835

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