Properties

Degree 1
Conductor 283
Sign $-0.543 + 0.839i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.991 − 0.133i)2-s + (0.166 + 0.986i)3-s + (0.964 + 0.264i)4-s + (0.944 + 0.328i)5-s + (−0.0334 − 0.999i)6-s + (0.784 + 0.619i)7-s + (−0.920 − 0.390i)8-s + (−0.944 + 0.328i)9-s + (−0.892 − 0.451i)10-s + (0.964 − 0.264i)11-s + (−0.100 + 0.994i)12-s + (−0.0334 − 0.999i)13-s + (−0.695 − 0.718i)14-s + (−0.166 + 0.986i)15-s + (0.860 + 0.509i)16-s + (−0.695 + 0.718i)17-s + ⋯
L(s,χ)  = 1  + (−0.991 − 0.133i)2-s + (0.166 + 0.986i)3-s + (0.964 + 0.264i)4-s + (0.944 + 0.328i)5-s + (−0.0334 − 0.999i)6-s + (0.784 + 0.619i)7-s + (−0.920 − 0.390i)8-s + (−0.944 + 0.328i)9-s + (−0.892 − 0.451i)10-s + (0.964 − 0.264i)11-s + (−0.100 + 0.994i)12-s + (−0.0334 − 0.999i)13-s + (−0.695 − 0.718i)14-s + (−0.166 + 0.986i)15-s + (0.860 + 0.509i)16-s + (−0.695 + 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.543 + 0.839i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.543 + 0.839i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $-0.543 + 0.839i$
motivic weight  =  \(0\)
character  :  $\chi_{283} (167, \cdot )$
Sato-Tate  :  $\mu(94)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 283,\ (1:\ ),\ -0.543 + 0.839i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.7625206310 + 1.401699473i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.7625206310 + 1.401699473i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8506357002 + 0.4698445584i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8506357002 + 0.4698445584i\)

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

−24.84849460714694950961343715514, −24.4946376826108802953918042954, −23.77846977167376009052653338760, −22.342281026976421493337521770436, −21.02000975080286333032340126129, −20.2879791001323191018355236332, −19.539897819315811091317811266009, −18.450776654728218785107304479100, −17.63504839241679812952450793194, −17.2193678499019753850116719027, −16.24772148802702232601012102243, −14.55484582651678284422273750696, −14.09723473033015119317833581628, −12.91239646220708975912117374055, −11.669819888986883214188660418545, −11.01741950439607689700201778912, −9.45708555021183904284465515780, −8.97969422773560398698234411679, −7.77383447934355283813056007200, −6.830087806114661396906846371261, −6.145991180040111008560173937242, −4.57747143684091462970477011212, −2.4514817922058435730972044882, −1.66782061704955853349645853897, −0.66255143589311791659923169174, 1.49455012657445917391134366497, 2.57576963943311204636830842952, 3.79747394813905279258866193726, 5.50968172655711549670660800257, 6.20320078702582036658581606171, 7.87101264286503992908679567595, 8.76318561982018438799381337119, 9.55048189090432263884443808669, 10.44573983589408953766569518180, 11.17998884298212549291952536022, 12.23997010494466585415730798463, 13.85819786084490418159025979538, 14.89486153949247384001252452762, 15.45545756256079268580145995952, 16.78583046182760842022196519880, 17.40243304664861431760590571705, 18.157484038837135330147341024369, 19.3178219652472872249278251785, 20.29389350422874973953550083409, 21.08892735838267094058923378891, 21.7931527987405854439178175604, 22.507686477739561319387290091627, 24.341542494268587660389224236790, 25.10653980510247496207202669309, 25.70420922709414374700084778699

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