Properties

Degree 1
Conductor 83
Sign $0.215 + 0.976i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.896 + 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.215 + 0.976i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (64, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.215 + 0.976i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.210325509 + 0.9722250393i$
$L(\frac12,\chi)$  $\approx$  $1.210325509 + 0.9722250393i$
$L(\chi,1)$  $\approx$  1.390214650 + 0.7466535796i
$L(1,\chi)$  $\approx$  1.390214650 + 0.7466535796i

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

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

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