Properties

Degree 1
Conductor 569
Sign $-0.999 + 0.0329i$
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 + 0.132i)2-s + (−0.110 + 0.993i)3-s + (0.964 − 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.0221 − 0.999i)6-s + (−0.839 + 0.544i)7-s + (−0.921 + 0.387i)8-s + (−0.975 − 0.219i)9-s + (−0.367 − 0.930i)10-s + (0.633 + 0.773i)11-s + (0.154 + 0.988i)12-s + (0.487 + 0.873i)13-s + (0.759 − 0.650i)14-s + (−0.991 + 0.132i)15-s + (0.862 − 0.506i)16-s + (0.984 + 0.176i)17-s + ⋯
L(s,χ)  = 1  + (−0.991 + 0.132i)2-s + (−0.110 + 0.993i)3-s + (0.964 − 0.262i)4-s + (0.240 + 0.970i)5-s + (−0.0221 − 0.999i)6-s + (−0.839 + 0.544i)7-s + (−0.921 + 0.387i)8-s + (−0.975 − 0.219i)9-s + (−0.367 − 0.930i)10-s + (0.633 + 0.773i)11-s + (0.154 + 0.988i)12-s + (0.487 + 0.873i)13-s + (0.759 − 0.650i)14-s + (−0.991 + 0.132i)15-s + (0.862 − 0.506i)16-s + (0.984 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.0329i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.0329i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.999 + 0.0329i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (111, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.999 + 0.0329i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.01316337233 + 0.7995618603i$
$L(\frac12,\chi)$  $\approx$  $0.01316337233 + 0.7995618603i$
$L(\chi,1)$  $\approx$  0.4872550023 + 0.4995688328i
$L(1,\chi)$  $\approx$  0.4872550023 + 0.4995688328i

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

−23.00699448491498288313050444149, −22.01038670710826336471402072823, −20.69425025209163142107932075684, −20.18401845983050885291118195158, −19.44020040324195073956024284980, −18.727734913274385450867179551729, −17.81833420763141774575611350355, −17.02941451689032540076342500258, −16.46443444980762431012366417100, −15.71476788356097352502199423958, −14.07118566490988435511833784717, −13.386514997171146270924151784689, −12.35462300435037039041299792570, −11.925501483970395818193015138658, −10.68206677044769405514972043370, −9.80353970246845962980241210368, −8.78207784021472955701215376366, −8.15989347876938946090693559655, −7.19985996651197473979912693469, −6.23626219021327986182952343371, −5.537184642395168587458451366661, −3.625777010731456374340102263349, −2.65489315198380504550519213098, −1.1426400272858700595891544051, −0.69780597367433666109889868563, 1.65929488730485397821066902531, 2.96805465275315488127513552843, 3.63620494084825840936121213745, 5.31956400715020079945397751349, 6.29565298758599493937610023903, 6.89103631767669206109106158820, 8.20513550624339820853431125567, 9.28950642024320818745778018597, 9.87085668096759542864560323341, 10.342185562920377155667030717887, 11.70577064327325375708192490154, 11.92599567567697807455780065124, 13.88529651648628574381486190190, 14.63963641591846742731991299668, 15.52705191878169835220481859364, 16.048690381925966834981765540884, 16.98158085905146210009609008137, 17.7757766243449622118008024392, 18.663133718385094626898961781231, 19.392844783750205172472170023241, 20.2005240963342887613363902308, 21.283376043149709769895594755112, 21.81795462631433379658311063603, 22.8136533625728163151600751585, 23.45474107921842746191955254499

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