Properties

Degree 1
Conductor 83
Sign $-0.576 - 0.817i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.543 + 0.839i)2-s + (−0.859 + 0.511i)3-s + (−0.409 + 0.912i)4-s + (−0.190 + 0.981i)5-s + (−0.896 − 0.443i)6-s + (−0.114 + 0.993i)7-s + (−0.988 + 0.152i)8-s + (0.477 − 0.878i)9-s + (−0.927 + 0.373i)10-s + (0.338 − 0.941i)11-s + (−0.114 − 0.993i)12-s + (0.264 + 0.964i)13-s + (−0.896 + 0.443i)14-s + (−0.338 − 0.941i)15-s + (−0.665 − 0.746i)16-s + (0.720 − 0.693i)17-s + ⋯
L(s,χ)  = 1  + (0.543 + 0.839i)2-s + (−0.859 + 0.511i)3-s + (−0.409 + 0.912i)4-s + (−0.190 + 0.981i)5-s + (−0.896 − 0.443i)6-s + (−0.114 + 0.993i)7-s + (−0.988 + 0.152i)8-s + (0.477 − 0.878i)9-s + (−0.927 + 0.373i)10-s + (0.338 − 0.941i)11-s + (−0.114 − 0.993i)12-s + (0.264 + 0.964i)13-s + (−0.896 + 0.443i)14-s + (−0.338 − 0.941i)15-s + (−0.665 − 0.746i)16-s + (0.720 − 0.693i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.576 - 0.817i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.576 - 0.817i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\n\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.576 - 0.817i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (58, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.576 - 0.817i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.4551532620 + 0.8776858444i$
$L(\frac12,\chi)$  $\approx$  $-0.4551532620 + 0.8776858444i$
$L(\chi,1)$  $\approx$  0.4611059991 + 0.7673110415i
$L(1,\chi)$  $\approx$  0.4611059991 + 0.7673110415i

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

−30.06526887991341762301555650053, −28.92477787430880056934155667436, −27.99056040521872477786995090444, −27.45564596228053725197562660029, −25.386687805093675738070178521750, −24.049072638117147715186791989658, −23.39280218673882134850799191023, −22.61233711079774147790156334434, −21.242799673207801482647424836430, −20.12219841878495651360928671473, −19.400169143193147466139270946804, −17.81771739069568260124965638616, −17.00737317045454962383136500729, −15.56457877775484835420456543912, −13.897807706719633056910319732840, −12.73578955364553252461332191720, −12.28843247515702161187743689674, −10.83305658674484815601141496504, −9.91370694395436034552892149919, −8.02634826742450422815149872209, −6.36113444313782304178735696956, −5.01509795686634142941036604313, −3.98826886316306961533431510150, −1.649393736231175984032570928398, −0.450560503932339680259795567343, 3.08794753022772968542513205019, 4.44254404672995969780079614219, 6.02051655400350031873893812032, 6.50463139497754647049413004891, 8.29783102514468349085150112670, 9.75720091901720440352509365851, 11.451656590378328334683430785661, 12.08395999402862251741319438079, 13.87398879352626126018120126577, 14.92658627062568620725983224905, 15.89992873144920021956673542415, 16.743928245443195571129564334351, 18.12939844269677671880977144029, 18.89950997152329734967016410264, 21.36559506589287494154821645775, 21.72778522954091171476267967922, 22.80411940388388124479288640564, 23.58206964082758358753879465935, 24.82229958280900917072252075087, 26.04006754427309385734248951990, 26.92905302538214441797396386091, 27.8823594095739659188577533271, 29.33960462384942223986960622940, 30.32367489288042116962253716640, 31.62550725228256182068985018552

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