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Properties

Label 1-83-83.36-r0-0-0
Degree $1$
Conductor $83$
Sign $0.999 + 0.0227i$
Analytic cond. $0.385450$
Root an. cond. $0.385450$
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  + (0.190 − 0.981i)2-s + (0.338 + 0.941i)3-s + (−0.927 − 0.373i)4-s + (0.896 + 0.443i)5-s + (0.988 − 0.152i)6-s + (0.0383 + 0.999i)7-s + (−0.543 + 0.839i)8-s + (−0.771 + 0.636i)9-s + (0.606 − 0.795i)10-s + (−0.114 − 0.993i)11-s + (0.0383 − 0.999i)12-s + (0.817 + 0.575i)13-s + (0.988 + 0.152i)14-s + (−0.114 + 0.993i)15-s + (0.720 + 0.693i)16-s + (−0.264 − 0.964i)17-s + ⋯
L(s)  = 1  + (0.190 − 0.981i)2-s + (0.338 + 0.941i)3-s + (−0.927 − 0.373i)4-s + (0.896 + 0.443i)5-s + (0.988 − 0.152i)6-s + (0.0383 + 0.999i)7-s + (−0.543 + 0.839i)8-s + (−0.771 + 0.636i)9-s + (0.606 − 0.795i)10-s + (−0.114 − 0.993i)11-s + (0.0383 − 0.999i)12-s + (0.817 + 0.575i)13-s + (0.988 + 0.152i)14-s + (−0.114 + 0.993i)15-s + (0.720 + 0.693i)16-s + (−0.264 − 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.999 + 0.0227i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.999 + 0.0227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.176309544 + 0.01340959859i\)
\(L(\frac12)\) \(\approx\) \(1.176309544 + 0.01340959859i\)
\(L(1)\) \(\approx\) \(1.217526970 - 0.09497473264i\)
\(L(1)\) \(\approx\) \(1.217526970 - 0.09497473264i\)

Euler product

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

−30.84638561547714796251848106720, −30.09720933618807788601050423338, −28.888581946407723197928371470915, −27.57731488925374853413256218112, −25.97628850660542049911633202804, −25.59460985748040634521417719796, −24.5628007998912128955615166686, −23.54061343616259750707479243102, −22.8244892559676728220769752542, −21.13258280716736090007126974802, −20.1067462506564030825440583665, −18.56792989215629255363863487645, −17.46586710849797767711561835311, −17.02356509671099726496534124647, −15.288093934881134427817245560925, −14.08436944511155180848873613771, −13.26707217955935204058031037006, −12.53749406574827228835976168412, −10.264541439145056056944073393471, −8.86207441638514993887943213191, −7.748040855353291612736652670844, −6.644516108825757095748250129779, −5.51603030301631574240475159253, −3.79605163075521204281962155807, −1.53263467860346248456257173935, 2.271205020565783964851205731866, 3.25133058764400011952221276564, 4.917537828178789125536608485812, 6.01643883965835553949507792663, 8.816094093028631317652598186946, 9.24748453466129511295523383514, 10.71567187513571573521720772526, 11.4009466371546583754355298832, 13.21201428993492447140410102079, 14.10863873575136785531997438562, 15.15542411691517451758805256140, 16.606579887493608496439803618126, 18.19052017049407749046869060861, 18.92306641390870295543506131484, 20.37643127215377282629577395097, 21.41756748610467376146763736910, 21.79313284450904075855572555508, 22.82053800096008279176671568293, 24.541590035127746200998014411921, 25.89616052646037734536712697304, 26.710470878410839388697368332060, 27.93984863109254323881015121991, 28.68959409372909861878020085347, 29.750458550852099092147109848, 31.00815130271046548834219744844

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