Properties

Degree 1
Conductor 569
Sign $0.0537 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.448 + 0.894i)2-s + (−0.921 − 0.387i)3-s + (−0.598 − 0.801i)4-s + (0.0663 − 0.997i)5-s + (0.759 − 0.650i)6-s + (0.903 + 0.428i)7-s + (0.984 − 0.176i)8-s + (0.699 + 0.714i)9-s + (0.862 + 0.506i)10-s + (−0.999 − 0.0442i)11-s + (0.240 + 0.970i)12-s + (−0.839 + 0.544i)13-s + (−0.787 + 0.616i)14-s + (−0.448 + 0.894i)15-s + (−0.283 + 0.958i)16-s + (0.814 − 0.580i)17-s + ⋯
L(s,χ)  = 1  + (−0.448 + 0.894i)2-s + (−0.921 − 0.387i)3-s + (−0.598 − 0.801i)4-s + (0.0663 − 0.997i)5-s + (0.759 − 0.650i)6-s + (0.903 + 0.428i)7-s + (0.984 − 0.176i)8-s + (0.699 + 0.714i)9-s + (0.862 + 0.506i)10-s + (−0.999 − 0.0442i)11-s + (0.240 + 0.970i)12-s + (−0.839 + 0.544i)13-s + (−0.787 + 0.616i)14-s + (−0.448 + 0.894i)15-s + (−0.283 + 0.958i)16-s + (0.814 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0537 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0537 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.0537 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (125, \cdot )$
Sato-Tate  :  $\mu(71)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.0537 + 0.998i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4326218145 + 0.4099659073i$
$L(\frac12,\chi)$  $\approx$  $0.4326218145 + 0.4099659073i$
$L(\chi,1)$  $\approx$  0.5847628937 + 0.1683441181i
$L(1,\chi)$  $\approx$  0.5847628937 + 0.1683441181i

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

−23.045834531296577152105628369156, −21.944508666029970804636178136994, −21.47937498684653378110520011550, −20.7644619801221235536037950127, −19.70509879552760907155989154807, −18.71884700955682397519592494011, −17.98127416144991194696478831277, −17.47706523484645727856066806779, −16.71768621914687228238809766327, −15.459748959814983665491611493033, −14.64589817207247467819490184503, −13.55204477723385665454075422153, −12.49827211754978220360241738217, −11.71003090949786954867045425361, −10.80549793225374010097161411846, −10.39562024673463292675924537624, −9.754565855821054488121411378848, −8.13785135254841955786957738875, −7.5376428068273644791361834786, −6.302419176988919369956911729631, −5.01485708271747916716603531083, −4.3030604197093133784267313484, −3.04811797471700861121106352159, −2.04573730291751379811292142825, −0.486106921438587042800515019911, 1.07337998866384964940401922352, 2.03866434588933205561835879465, 4.4554409474270056113448202265, 5.136041385839804787355530897304, 5.63010273728507545890037353650, 6.808572436731710216822184421638, 7.94934090460488358454123922165, 8.25361522426475895991626014439, 9.66335384675499545848745342826, 10.32800698025369215143565779514, 11.63730513612199143820809581019, 12.287507765353141013497837367905, 13.33021108422531699389255172036, 14.18456106073819317169592558874, 15.28777896728221798504326670800, 16.17866435046403801893952507787, 16.75391430986400023997026886517, 17.51284473445027132401682146109, 18.23288541983015431071835467811, 18.89875259920786906134957805321, 19.95993814462210149580023653948, 21.21668623794632931598120975936, 21.751786652084233675391049380069, 23.164549923831920707387062082905, 23.584563800107075448426390123468

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