Properties

Degree 1
Conductor 569
Sign $0.991 + 0.133i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.787 + 0.616i)2-s + (0.525 − 0.850i)3-s + (0.240 − 0.970i)4-s + (0.937 + 0.346i)5-s + (0.110 + 0.993i)6-s + (0.964 + 0.262i)7-s + (0.408 + 0.912i)8-s + (−0.448 − 0.894i)9-s + (−0.952 + 0.304i)10-s + (0.283 + 0.958i)11-s + (−0.699 − 0.714i)12-s + (0.562 − 0.826i)13-s + (−0.921 + 0.387i)14-s + (0.787 − 0.616i)15-s + (−0.883 − 0.467i)16-s + (0.633 + 0.773i)17-s + ⋯
L(s,χ)  = 1  + (−0.787 + 0.616i)2-s + (0.525 − 0.850i)3-s + (0.240 − 0.970i)4-s + (0.937 + 0.346i)5-s + (0.110 + 0.993i)6-s + (0.964 + 0.262i)7-s + (0.408 + 0.912i)8-s + (−0.448 − 0.894i)9-s + (−0.952 + 0.304i)10-s + (0.283 + 0.958i)11-s + (−0.699 − 0.714i)12-s + (0.562 − 0.826i)13-s + (−0.921 + 0.387i)14-s + (0.787 − 0.616i)15-s + (−0.883 − 0.467i)16-s + (0.633 + 0.773i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.991 + 0.133i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.991 + 0.133i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.991 + 0.133i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (100, \cdot )$
Sato-Tate  :  $\mu(142)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.991 + 0.133i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.567830363 + 0.1048682445i$
$L(\frac12,\chi)$  $\approx$  $1.567830363 + 0.1048682445i$
$L(\chi,1)$  $\approx$  1.170414374 + 0.06523149867i
$L(1,\chi)$  $\approx$  1.170414374 + 0.06523149867i

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

−22.99904809930485878314727829933, −21.9212828151964497608958833447, −21.10933405463169029582140653204, −20.9500936240930894631034207655, −20.20775650584352037486593344089, −18.91809042694360465328245902637, −18.49757345594329024093835274285, −17.10152201334403402025633523083, −16.81104396864240291102470399625, −16.01350793097832821230250535287, −14.617520126976163820718563270253, −13.91183732849641421253868307738, −13.18167785584337282786622201192, −11.68519921545106775899500644392, −11.13597257459809431584060102070, −10.11289718182437884088210246689, −9.53210388096386385174290022016, −8.49276232860725587152175991605, −8.18316305235372017563512917030, −6.64204443079086481287141723, −5.32480126108038707148690539348, −4.291734498888006374973714771362, −3.30907582395651302551479926082, −2.15477560394208638049818758923, −1.234621665881230136956584495569, 1.382168758767326666892234686275, 1.827015635614745863886239370986, 3.08538636289299660374923160056, 5.019068291345261749265507938819, 5.83477523384172269488669367971, 6.84686493306139961724179027011, 7.489155418791829412629285869217, 8.53998655686008823443183545807, 9.10334077575975147755882350696, 10.2221158105485887394993487065, 11.02952148712702070733385428707, 12.23541099405656061895836243380, 13.321895642045640476572765126742, 14.116444840993244900878517333013, 14.95430698763089244665193959580, 15.33673019992186713006246620120, 17.12441113786316333840817863271, 17.511764482889414649766651361522, 18.11751449308427556903585261653, 18.83537327317396461924836694413, 19.80182277455529660647220939489, 20.58569358912054046098163724274, 21.387428154625284018207331888500, 22.75555788376246220828821518059, 23.56191713925577453103180675792

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