Properties

Degree 1
Conductor 569
Sign $0.0251 + 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.862 + 0.506i)2-s + (0.903 − 0.428i)3-s + (0.487 + 0.873i)4-s + (0.562 + 0.826i)5-s + (0.996 + 0.0883i)6-s + (−0.666 + 0.745i)7-s + (−0.0221 + 0.999i)8-s + (0.633 − 0.773i)9-s + (0.0663 + 0.997i)10-s + (−0.921 + 0.387i)11-s + (0.814 + 0.580i)12-s + (−0.448 + 0.894i)13-s + (−0.952 + 0.304i)14-s + (0.862 + 0.506i)15-s + (−0.525 + 0.850i)16-s + (0.759 − 0.650i)17-s + ⋯
L(s,χ)  = 1  + (0.862 + 0.506i)2-s + (0.903 − 0.428i)3-s + (0.487 + 0.873i)4-s + (0.562 + 0.826i)5-s + (0.996 + 0.0883i)6-s + (−0.666 + 0.745i)7-s + (−0.0221 + 0.999i)8-s + (0.633 − 0.773i)9-s + (0.0663 + 0.997i)10-s + (−0.921 + 0.387i)11-s + (0.814 + 0.580i)12-s + (−0.448 + 0.894i)13-s + (−0.952 + 0.304i)14-s + (0.862 + 0.506i)15-s + (−0.525 + 0.850i)16-s + (0.759 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0251 + 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0251 + 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.0251 + 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (107, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.0251 + 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.130441570 + 2.184701372i$
$L(\frac12,\chi)$  $\approx$  $2.130441570 + 2.184701372i$
$L(\chi,1)$  $\approx$  1.942171343 + 1.014481101i
$L(1,\chi)$  $\approx$  1.942171343 + 1.014481101i

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

−23.02035318962374935985878116739, −21.96956545625830582811691417919, −21.23519267709695238894788081886, −20.69106481764509623606143840515, −19.89960338752260966355383414568, −19.40464355061648943946050433441, −18.26578591010740183728122393893, −16.87139631462790934755456962504, −16.07761762849234301373719541169, −15.41264567329894640392962138089, −14.26251689375542422940608873643, −13.6876854736324257715619866689, −12.92668932729017657376826911600, −12.398004773057206450709940377972, −10.79262909569040299976329304270, −10.01230862048843995492820114646, −9.65345984829970679203819665589, −8.223569543886583672940984999, −7.38657553338428664094458971939, −5.81279405454378778221924695918, −5.242963980387169710654747845432, −4.02702435557557330859815018944, −3.322538038171682254411265259225, −2.307939780494263752117124130297, −1.085289906756322619732027647345, 2.17910005001924256715661811831, 2.657020363711479570535056057962, 3.49290639813164620255019971096, 4.90204950849012286535276583712, 5.906774028025647642382174111427, 7.00934195649182306445515528516, 7.26282561712186852827951262254, 8.63716916047017953805341551019, 9.47127554368076881997885422905, 10.53391034855273848958947264514, 11.98333025164083667501778291150, 12.535734051686450115839422742580, 13.575945014682626274718711464915, 14.06834212230295297951942998876, 14.91776966959863650620862420243, 15.59020789337402084052044239426, 16.45249970638957465088413150298, 17.79918868724612287916160767187, 18.45355502343853052582764183340, 19.22992701797128009555445082504, 20.373608109949598350435026768681, 21.12437676358398841003698699956, 21.91174086924703007468055015111, 22.54231053024340436610332537707, 23.6003591960000262048964053509

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