Properties

Degree $1$
Conductor $283$
Sign $-0.960 + 0.277i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.480 + 0.876i)2-s + (−0.231 − 0.972i)3-s + (−0.538 − 0.842i)4-s + (0.892 + 0.451i)5-s + (0.964 + 0.264i)6-s + (0.593 + 0.805i)7-s + (0.997 − 0.0667i)8-s + (−0.892 + 0.451i)9-s + (−0.824 + 0.565i)10-s + (−0.538 + 0.842i)11-s + (−0.695 + 0.718i)12-s + (0.964 + 0.264i)13-s + (−0.991 + 0.133i)14-s + (0.231 − 0.972i)15-s + (−0.420 + 0.907i)16-s + (−0.991 − 0.133i)17-s + ⋯
L(s,χ)  = 1  + (−0.480 + 0.876i)2-s + (−0.231 − 0.972i)3-s + (−0.538 − 0.842i)4-s + (0.892 + 0.451i)5-s + (0.964 + 0.264i)6-s + (0.593 + 0.805i)7-s + (0.997 − 0.0667i)8-s + (−0.892 + 0.451i)9-s + (−0.824 + 0.565i)10-s + (−0.538 + 0.842i)11-s + (−0.695 + 0.718i)12-s + (0.964 + 0.264i)13-s + (−0.991 + 0.133i)14-s + (0.231 − 0.972i)15-s + (−0.420 + 0.907i)16-s + (−0.991 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.960 + 0.277i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.960 + 0.277i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.960 + 0.277i$
Motivic weight: \(0\)
Character: $\chi_{283} (172, \cdot )$
Sato-Tate group: $\mu(94)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.960 + 0.277i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.1056489219 + 0.7450943056i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.1056489219 + 0.7450943056i\)
\(L(\chi,1)\) \(\approx\) \(0.7041314312 + 0.2946961245i\)
\(L(1,\chi)\) \(\approx\) \(0.7041314312 + 0.2946961245i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.29098895475170258253860175643, −23.86794411717478181205139809796, −22.90924686507142147769629889266, −21.72369345199028957318224490461, −21.3047834897000850332276396568, −20.505423716533315089344586527521, −19.89506970772366629026592773452, −18.30169816725416613260696045898, −17.70705627464611345663622381874, −16.69543839894952878715586063996, −16.22222543973660792325599610330, −14.58107576087223646549037661059, −13.589110396213759984827106545664, −12.83165045806968137852056304060, −11.31643069184110700153948012348, −10.66744345570747041986430030329, −10.11312806149466127349942260841, −8.69729167609919203699158051238, −8.431147189984731502167475842812, −6.42729965601521104802237963838, −5.08234069571699920553570643105, −4.23173907382472009538875485309, −3.060581836109935037543071204255, −1.60132837488018991781450554631, −0.27071314834287550810205206812, 1.59719311122375447558222128254, 2.313730298411449540608035836735, 4.72507411753400952004014776313, 5.846609565054200424122862296242, 6.41621976521069788812422925290, 7.49335695119043213322639056525, 8.47226465269059624573460480339, 9.39107770808438747275044543884, 10.6666885324634022687307460978, 11.59331188536151059035695941100, 13.19252486342622899207569158483, 13.57207243338615509783417542599, 14.843176382195216035144383900266, 15.46492824525794109397494743017, 16.97554720321952765005321009040, 17.65260007768935300557903821210, 18.31739124045240463841747722989, 18.80387744376231317528657122233, 20.09290432170308142884028724992, 21.38616575389158336910356217267, 22.43671657475957861589639060912, 23.33220495172939540158248084917, 24.10817007805801112961114148401, 25.024485252202651177735185158178, 25.59190309439493848931812823922

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