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Properties

Label 1-83-83.15-r1-0-0
Degree $1$
Conductor $83$
Sign $0.717 - 0.696i$
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

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Normalization:  

Dirichlet series

L(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)  = 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.717 - 0.696i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 0.717 - 0.696i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.595667274 - 0.6474517247i\)
\(L(\frac12)\) \(\approx\) \(1.595667274 - 0.6474517247i\)
\(L(1)\) \(\approx\) \(1.259750130 + 6.812072557\times10^{-5}i\)
\(L(1)\) \(\approx\) \(1.259750130 + 6.812072557\times10^{-5}i\)

Euler product

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

−30.75097889529016326378588501964, −30.105683592535342143446003668646, −28.461174181104155349197642077593, −27.670241262370228235018120573871, −26.685261751667275498858547735167, −25.673946146201363994024036636579, −24.27226756699971166930769298258, −22.75069727497862423666206002078, −22.108932883006824305511989094601, −20.97767548817273180436633324863, −19.80054185962937820770260314291, −19.18079865715138316452309178151, −18.24365992727228019804294493274, −16.093217156650215462861288077349, −15.00708812029135893939908479930, −14.17993506822743380672334521541, −12.79315716706627952329701376178, −11.64032445848766176787931168863, −10.44967715561871143366273636727, −9.20492135937215362237021161449, −8.27876686874903392153557989746, −6.23371987980146872136129683893, −4.16188497496155349041026979468, −3.446286733957844905298117015341, −2.01539443166312964080920955799, 0.71156283135180305076566938372, 3.48838057411678593268803610585, 4.32272581495287886849887508848, 6.4357284378576562746781694846, 7.416017017669388085390513076064, 8.524421769970617876759924479303, 9.4459541052756703770938148743, 11.7205502274000801994056341495, 13.18606641394945848195593566837, 13.67250136106992769407100665755, 15.12080925575182779920028133367, 15.98110562349773273077401109088, 17.1105615096714840592132294312, 18.49009285563662774884346516420, 19.726305141144794446912430719947, 20.47627165099413779875950617630, 22.18393419340668735182583097496, 23.379345880921963801925222626113, 24.12350247512524014393929936797, 25.06832414714613295144920115247, 26.12896997845136829028226083338, 26.91208459537349704238988782484, 28.04987414314212981444184005536, 29.92802489961568696642034642884, 30.66175863903993855239151279242

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