Properties

Degree 1
Conductor 1021
Sign $0.588 - 0.808i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.542 − 0.840i)2-s + (−0.850 − 0.526i)3-s + (−0.412 + 0.911i)4-s + (0.0184 + 0.999i)5-s + (0.0184 + 0.999i)6-s + (0.687 + 0.726i)7-s + (0.989 − 0.147i)8-s + (0.445 + 0.895i)9-s + (0.830 − 0.557i)10-s + (−0.999 + 0.0369i)11-s + (0.830 − 0.557i)12-s + (0.932 − 0.361i)13-s + (0.237 − 0.971i)14-s + (0.510 − 0.859i)15-s + (−0.659 − 0.751i)16-s + (0.572 − 0.819i)17-s + ⋯
L(s,χ)  = 1  + (−0.542 − 0.840i)2-s + (−0.850 − 0.526i)3-s + (−0.412 + 0.911i)4-s + (0.0184 + 0.999i)5-s + (0.0184 + 0.999i)6-s + (0.687 + 0.726i)7-s + (0.989 − 0.147i)8-s + (0.445 + 0.895i)9-s + (0.830 − 0.557i)10-s + (−0.999 + 0.0369i)11-s + (0.830 − 0.557i)12-s + (0.932 − 0.361i)13-s + (0.237 − 0.971i)14-s + (0.510 − 0.859i)15-s + (−0.659 − 0.751i)16-s + (0.572 − 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.588 - 0.808i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.588 - 0.808i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1021\)
\( \varepsilon \)  =  $0.588 - 0.808i$
motivic weight  =  \(0\)
character  :  $\chi_{1021} (235, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ 0.588 - 0.808i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7312945404 - 0.3724479828i$
$L(\frac12,\chi)$  $\approx$  $0.7312945404 - 0.3724479828i$
$L(\chi,1)$  $\approx$  0.6473109235 - 0.2040552155i
$L(1,\chi)$  $\approx$  0.6473109235 - 0.2040552155i

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

−21.61704463748516832169948839636, −20.85781995777235071674905605358, −20.37182653462225282908719281674, −19.2198999047183468784124580128, −18.04494304564736763676341787335, −17.866609531210516769636374975926, −16.80777891265049023581716633705, −16.241815902920758360349672241934, −15.969038620098620883052926281617, −14.75921010565460555131620285369, −14.033069871935626885401726451022, −13.01068950663782596536753698605, −12.18235447314932701102752165874, −10.94937129790541428446900551854, −10.488461255751956603117111766779, −9.68171716181106593707178174200, −8.60815198029280252248561119861, −8.04856823937856356903719639273, −7.076631083801960999568653457215, −5.87419554086962573339455066556, −5.462230630784773370048480422163, −4.4997609536079115660892450133, −3.87725677368856235574262470337, −1.588490753848174076958377769947, −0.876489759166555451565161795065, 0.68047375133938586938135525692, 2.00818141810951246101250683769, 2.61056507718363286373870487060, 3.73578659080032385750590567157, 5.10974426955303922419439253723, 5.72080821872628596024252808715, 7.02786194658631817932726442380, 7.684076741792382457023822116999, 8.43068361823741578846545808496, 9.690114438429438813281617227198, 10.48317864641207418166249613687, 11.20658828820347430481108604404, 11.63655063611925269203048748709, 12.46350705300690181060805082064, 13.48974866596344317864777660456, 13.98189096162734231642467696158, 15.54578631899973773334644647659, 15.92035201861164071703566755846, 17.30208006869539077405550099589, 17.838981751122897361422513769820, 18.56367448194621169191874678129, 18.62188906625086935302217127256, 19.775530727583694531913099713492, 20.85353908494932445478214334116, 21.4615390930817693435457491820

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