Properties

Degree 1
Conductor 1021
Sign $-0.875 - 0.483i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.903 + 0.429i)2-s + (−0.273 − 0.961i)3-s + (0.631 + 0.775i)4-s + (0.165 − 0.986i)5-s + (0.165 − 0.986i)6-s + (0.510 − 0.859i)7-s + (0.237 + 0.971i)8-s + (−0.850 + 0.526i)9-s + (0.572 − 0.819i)10-s + (−0.945 − 0.326i)11-s + (0.572 − 0.819i)12-s + (−0.982 − 0.183i)13-s + (0.830 − 0.557i)14-s + (−0.993 + 0.110i)15-s + (−0.201 + 0.979i)16-s + (−0.713 + 0.700i)17-s + ⋯
L(s,χ)  = 1  + (0.903 + 0.429i)2-s + (−0.273 − 0.961i)3-s + (0.631 + 0.775i)4-s + (0.165 − 0.986i)5-s + (0.165 − 0.986i)6-s + (0.510 − 0.859i)7-s + (0.237 + 0.971i)8-s + (−0.850 + 0.526i)9-s + (0.572 − 0.819i)10-s + (−0.945 − 0.326i)11-s + (0.572 − 0.819i)12-s + (−0.982 − 0.183i)13-s + (0.830 − 0.557i)14-s + (−0.993 + 0.110i)15-s + (−0.201 + 0.979i)16-s + (−0.713 + 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.875 - 0.483i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.875 - 0.483i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $-0.875 - 0.483i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (255, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ -0.875 - 0.483i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3248320446 - 1.259866412i$
$L(\frac12,\chi)$  $\approx$  $0.3248320446 - 1.259866412i$
$L(\chi,1)$  $\approx$  1.203231001 - 0.4812203880i
$L(1,\chi)$  $\approx$  1.203231001 - 0.4812203880i

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

−21.93702152373165729385963467177, −21.24811709139732685302585156243, −20.84420780864206265153570972420, −19.7045399873913497599079581737, −18.97919299193833929008291052559, −18.00590735538121739275421810074, −17.3500346420996439555507201330, −15.93306055629042920152269004359, −15.49856825420784464074285302022, −14.81397375084001933842980249182, −14.26898954808941494170890134016, −13.30155382098385372201065232399, −12.09347285643618018445978504894, −11.633790938621854926600520003503, −10.78817693949749404291552745733, −10.11077006027191482058598335737, −9.48272314364962976673614111894, −8.150432628263110172143091837677, −6.94451689190706818266704070083, −6.1021059436981479848985157350, −5.16470086415002092664976981661, −4.7206776541241294934542938322, −3.53503572765935865856876744206, −2.628232458789842108518037952357, −2.06505764275892898422671813269, 0.360768745376384456815343292585, 1.84315397274138788450631430751, 2.5490433456083525428087473397, 4.041255570641963393388407027494, 4.91381876084814663282583609171, 5.443100000680319001959332588387, 6.57094733909368617561997094131, 7.2212856431275331138744915796, 8.259189024128130776271539884644, 8.51101868162269924594364812035, 10.37358907698044864302594520899, 11.04046918577041119563297875879, 12.141661097421562467591607592051, 12.64188335910459035955960128353, 13.383824648909233397896504667, 13.84669498596880532869434018168, 14.84252407135165206608434116242, 15.76053941217102606522045214215, 16.84061376970625808484878069983, 17.12654350173768564170033058093, 17.79060658212534509095306259543, 19.01297445426727167442662133946, 20.09992906663957547682221092614, 20.3436914299717763184610001144, 21.41886411655703619571396826552

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