Properties

Degree 1
Conductor 83
Sign $0.110 - 0.993i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.264 − 0.964i)2-s + (0.896 − 0.443i)3-s + (−0.859 + 0.511i)4-s + (0.817 + 0.575i)5-s + (−0.665 − 0.746i)6-s + (−0.543 − 0.839i)7-s + (0.720 + 0.693i)8-s + (0.606 − 0.795i)9-s + (0.338 − 0.941i)10-s + (0.988 − 0.152i)11-s + (−0.543 + 0.839i)12-s + (−0.973 + 0.227i)13-s + (−0.665 + 0.746i)14-s + (0.988 + 0.152i)15-s + (0.477 − 0.878i)16-s + (−0.771 − 0.636i)17-s + ⋯
L(s,χ)  = 1  + (−0.264 − 0.964i)2-s + (0.896 − 0.443i)3-s + (−0.859 + 0.511i)4-s + (0.817 + 0.575i)5-s + (−0.665 − 0.746i)6-s + (−0.543 − 0.839i)7-s + (0.720 + 0.693i)8-s + (0.606 − 0.795i)9-s + (0.338 − 0.941i)10-s + (0.988 − 0.152i)11-s + (−0.543 + 0.839i)12-s + (−0.973 + 0.227i)13-s + (−0.665 + 0.746i)14-s + (0.988 + 0.152i)15-s + (0.477 − 0.878i)16-s + (−0.771 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.110 - 0.993i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.110 - 0.993i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.110 - 0.993i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (68, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 83,\ (0:\ ),\ 0.110 - 0.993i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8272514179 - 0.7402696205i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8272514179 - 0.7402696205i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9876813389 - 0.5923542385i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9876813389 - 0.5923542385i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−31.66854388291532292847384522151, −30.27374776095882253291637471160, −28.58904045337261394086000186858, −27.83589600134007338609681033957, −26.565600921074225524258764475466, −25.70802648631954311648750643508, −24.83698235795432913635243342702, −24.30779633481576729028060810481, −22.16523972927309524218629466341, −21.86382229750459868943586730255, −20.06756075174487915835679757153, −19.24751172497369260711457211816, −17.80774117173996967296196668960, −16.7794294896382810362956074787, −15.63334781603482209327348295513, −14.73380150455148681405293160984, −13.65638786668493874162489136409, −12.57505199545611121177164101596, −10.12213751706785827444685591657, −9.218347831095779703467065395239, −8.602235019966608112314599621077, −6.91531460946387631832278232342, −5.55145163398802079192943972903, −4.25163962065369963403159610361, −2.173473203565586430042134098617, 1.62553865911495105582477057056, 2.90187044670249115916897179118, 4.12743098606249065695018973061, 6.53585357259614904378332740419, 7.79170590950574997109182778634, 9.485471283444910013789274785772, 9.86834280083972482822746422897, 11.51900157243408631633970864537, 12.91404188436052514969402201102, 13.8324797327684291691331295030, 14.5549973757824972121704260349, 16.76828668612312483642911340768, 17.8269406439776865766508442554, 18.92658721776966436290440349721, 19.78316653827055422029900883154, 20.63627801265905092169230089842, 21.91168068951241512071883658069, 22.74166233835319126463268733178, 24.414876284520624572893790761407, 25.61164321799407018058880405799, 26.4817078797961705151710245412, 27.21201697038739846317025409194, 29.029102493066292744009860276534, 29.69891811331788051091505471657, 30.23349498787291407307462524264

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