Properties

Degree 1
Conductor 569
Sign $-0.723 + 0.690i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.325 − 0.945i)2-s + (0.487 − 0.873i)3-s + (−0.787 − 0.616i)4-s + (0.984 − 0.176i)5-s + (−0.666 − 0.745i)6-s + (−0.991 + 0.132i)7-s + (−0.839 + 0.544i)8-s + (−0.525 − 0.850i)9-s + (0.154 − 0.988i)10-s + (−0.598 − 0.801i)11-s + (−0.921 + 0.387i)12-s + (−0.883 − 0.467i)13-s + (−0.197 + 0.980i)14-s + (0.325 − 0.945i)15-s + (0.240 + 0.970i)16-s + (0.903 − 0.428i)17-s + ⋯
L(s,χ)  = 1  + (0.325 − 0.945i)2-s + (0.487 − 0.873i)3-s + (−0.787 − 0.616i)4-s + (0.984 − 0.176i)5-s + (−0.666 − 0.745i)6-s + (−0.991 + 0.132i)7-s + (−0.839 + 0.544i)8-s + (−0.525 − 0.850i)9-s + (0.154 − 0.988i)10-s + (−0.598 − 0.801i)11-s + (−0.921 + 0.387i)12-s + (−0.883 − 0.467i)13-s + (−0.197 + 0.980i)14-s + (0.325 − 0.945i)15-s + (0.240 + 0.970i)16-s + (0.903 − 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.723 + 0.690i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.723 + 0.690i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $-0.723 + 0.690i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (219, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ -0.723 + 0.690i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.4613405263 - 1.151022978i$
$L(\frac12,\chi)$  $\approx$  $-0.4613405263 - 1.151022978i$
$L(\chi,1)$  $\approx$  0.5938794878 - 0.9888770047i
$L(1,\chi)$  $\approx$  0.5938794878 - 0.9888770047i

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

−23.69662223947160469561012176270, −22.876099462929338973107829421163, −22.067697427801298677616906187779, −21.514430655675593718716691727237, −20.803218880094603306937622008887, −19.5692159235085233721184077706, −18.76958137211404278603397545537, −17.40949091336132812013420510818, −17.07083401634942443756211924629, −16.08198181934552908876795440981, −15.31023436577401538434265802046, −14.61903026459625253861394468178, −13.74582446783597459306972039822, −13.11440972839543551103630258659, −12.13310188909465450108461689572, −10.390496222332650448244120384, −9.77252968444616518190367380731, −9.25457263273539985176075604291, −8.05645425139285425229701263748, −7.06448460354616195693062910163, −6.10881499688771983455145267059, −5.18737154037828777399680139971, −4.34282028625096695761704078709, −3.20361078080764954272199918761, −2.30064644447744450848779785774, 0.50233168390733643490441013141, 1.798091896129123516649511633335, 2.79451354231380367350550398361, 3.281114802744905853163656205428, 4.995400597840260301376726104713, 5.90332326252530772587156447160, 6.6593332855051239747356486443, 8.18330090561347236096094588394, 8.938792716195262548397041444440, 9.980570258533282133742121172964, 10.44576820548231901110664778762, 12.03038794698871240387150324794, 12.616296545225185672921926031498, 13.16332311614912099165496274530, 14.11035828249855111560981108162, 14.53324112674691033837266614675, 16.00897540004739162275296397134, 17.171723615427901200625986343890, 18.01755267067251951182502567818, 18.86244767534393012752109785256, 19.29375783189344663647808848275, 20.288439264836057560624777968565, 21.01163068401410707083026140049, 21.77864109339361753494586926629, 22.697339837690445868305341983451

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