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Properties

Label 1-83-83.48-r0-0-0
Degree $1$
Conductor $83$
Sign $0.215 - 0.976i$
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  + (0.896 − 0.443i)2-s + (−0.114 − 0.993i)3-s + (0.606 − 0.795i)4-s + (0.988 + 0.152i)5-s + (−0.543 − 0.839i)6-s + (−0.859 + 0.511i)7-s + (0.190 − 0.981i)8-s + (−0.973 + 0.227i)9-s + (0.953 − 0.301i)10-s + (0.0383 + 0.999i)11-s + (−0.859 − 0.511i)12-s + (−0.665 + 0.746i)13-s + (−0.543 + 0.839i)14-s + (0.0383 − 0.999i)15-s + (−0.264 − 0.964i)16-s + (0.817 − 0.575i)17-s + ⋯
L(s)  = 1  + (0.896 − 0.443i)2-s + (−0.114 − 0.993i)3-s + (0.606 − 0.795i)4-s + (0.988 + 0.152i)5-s + (−0.543 − 0.839i)6-s + (−0.859 + 0.511i)7-s + (0.190 − 0.981i)8-s + (−0.973 + 0.227i)9-s + (0.953 − 0.301i)10-s + (0.0383 + 0.999i)11-s + (−0.859 − 0.511i)12-s + (−0.665 + 0.746i)13-s + (−0.543 + 0.839i)14-s + (0.0383 − 0.999i)15-s + (−0.264 − 0.964i)16-s + (0.817 − 0.575i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.215 - 0.976i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.215 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210325509 - 0.9722250393i\)
\(L(\frac12)\) \(\approx\) \(1.210325509 - 0.9722250393i\)
\(L(1)\) \(\approx\) \(1.390214650 - 0.7466535796i\)
\(L(1)\) \(\approx\) \(1.390214650 - 0.7466535796i\)

Euler product

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

−31.67397368498509107456415213497, −29.687644426497891671440616739, −29.50201620507349478983833278136, −27.93258431729641389241985665305, −26.47632183493994599094309343597, −25.81105341077402888297164872290, −24.78259939476904881268265919966, −23.394483954560898396999113417390, −22.37958038315760524987414654858, −21.598869448851861296760842898298, −20.79726742558579106725888584416, −19.5307516483706958152385637767, −17.3118554072658791371397130507, −16.754768311931492382992182021437, −15.70310063162485977956817347710, −14.46128439213852765462417088043, −13.546748501676732854826107709484, −12.3597763777450430470854835654, −10.7152350938311077337724004762, −9.75919414960013065664538629223, −8.17549938798685068947092545353, −6.239372871248736777010585545787, −5.55665470677347318447305101937, −4.024789716371712963665649930569, −2.842258392168211012589619526632, 1.86642866695170087264852634125, 2.80683138854129499288533214730, 4.98323407790975826693987768195, 6.268465524429794567839900191597, 7.00114781398299664519251719382, 9.27397656106847561585120532442, 10.431325203202399883643979901286, 12.171628308582834814011681907284, 12.60762425364604575209108094206, 13.86093929537527909787021269088, 14.65932979199422868137632508091, 16.33875242319882638611228120051, 17.75068131128119875167920092170, 18.88811599684305295110902328570, 19.759756122306900131885418889348, 21.10609466813528796944753754270, 22.21283002616159661722311414461, 22.98649770671554126234235338678, 24.16586660819502528302530089405, 25.24187520589607537989705057322, 25.77496132196718014257856666860, 28.1374064828082497668904140349, 28.912834011084391814118310503571, 29.61935974485635733605957841889, 30.521889785272862428416603625332

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