↑

Properties

Label 1-83-83.73-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.576 + 0.817i$
Analytic cond. $8.91958$
Root an. cond. $8.91958$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.576 + 0.817i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.576 + 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4551532620 - 0.8776858444i\)
\(L(\frac12)\) \(\approx\) \(-0.4551532620 - 0.8776858444i\)
\(L(1)\) \(\approx\) \(0.4611059991 - 0.7673110415i\)
\(L(1)\) \(\approx\) \(0.4611059991 - 0.7673110415i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 + (0.543 - 0.839i)T \)
3 \( 1 + (-0.859 - 0.511i)T \)
5 \( 1 + (-0.190 - 0.981i)T \)
7 \( 1 + (-0.114 - 0.993i)T \)
11 \( 1 + (0.338 + 0.941i)T \)
13 \( 1 + (0.264 - 0.964i)T \)
17 \( 1 + (0.720 + 0.693i)T \)
19 \( 1 + (-0.953 - 0.301i)T \)
23 \( 1 + (-0.997 + 0.0765i)T \)
29 \( 1 + (0.817 + 0.575i)T \)
31 \( 1 + (0.988 - 0.152i)T \)
37 \( 1 + (0.477 - 0.878i)T \)
41 \( 1 + (-0.543 - 0.839i)T \)
43 \( 1 + (-0.0383 - 0.999i)T \)
47 \( 1 + (-0.606 + 0.795i)T \)
53 \( 1 + (-0.606 - 0.795i)T \)
59 \( 1 + (-0.771 - 0.636i)T \)
61 \( 1 + (-0.973 - 0.227i)T \)
67 \( 1 + (0.665 - 0.746i)T \)
71 \( 1 + (0.114 - 0.993i)T \)
73 \( 1 + (0.927 + 0.373i)T \)
79 \( 1 + (0.409 + 0.912i)T \)
89 \( 1 + (-0.896 + 0.443i)T \)
97 \( 1 + (-0.896 - 0.443i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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