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  + (0.237 + 0.971i)2-s + (0.739 + 0.673i)3-s + (−0.886 + 0.462i)4-s + (−0.478 + 0.878i)5-s + (−0.478 + 0.878i)6-s + (−0.999 − 0.0369i)7-s + (−0.659 − 0.751i)8-s + (0.0922 + 0.995i)9-s + (−0.966 − 0.255i)10-s + (−0.542 − 0.840i)11-s + (−0.966 − 0.255i)12-s + (−0.850 − 0.526i)13-s + (−0.201 − 0.979i)14-s + (−0.945 + 0.326i)15-s + (0.572 − 0.819i)16-s + (0.687 + 0.726i)17-s + ⋯
L(s,χ)  = 1  + (0.237 + 0.971i)2-s + (0.739 + 0.673i)3-s + (−0.886 + 0.462i)4-s + (−0.478 + 0.878i)5-s + (−0.478 + 0.878i)6-s + (−0.999 − 0.0369i)7-s + (−0.659 − 0.751i)8-s + (0.0922 + 0.995i)9-s + (−0.966 − 0.255i)10-s + (−0.542 − 0.840i)11-s + (−0.966 − 0.255i)12-s + (−0.850 − 0.526i)13-s + (−0.201 − 0.979i)14-s + (−0.945 + 0.326i)15-s + (0.572 − 0.819i)16-s + (0.687 + 0.726i)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  :  $\chi_{1021} (335, \cdot )$
Sato-Tate  :  $\mu(85)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1021,\ (0:\ ),\ 0.875 - 0.483i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1480844830 - 0.03818692090i$
$L(\frac12,\chi)$  $\approx$  $0.1480844830 - 0.03818692090i$
$L(\chi,1)$  $\approx$  0.5791893963 + 0.5986278037i
$L(1,\chi)$  $\approx$  0.5791893963 + 0.5986278037i

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

−21.33497733421633989971819529196, −20.57213165126606797594448920269, −20.25647050664430908433145728711, −19.33619038181089507961382617074, −18.902670312000611111675543528960, −18.16817508147408047432593554096, −17.02587945256755389504416128489, −16.21109275037959839808081702569, −15.05163983571492510466047297340, −14.45310100579422822446978998506, −13.43038346534062126500073333102, −12.71864896111359242249852386878, −12.392319963490638098145308988504, −11.683966286926266727373702434081, −10.27939693641714879074586207740, −9.43301855455422425279830175736, −9.10395304094016208313227133425, −7.88543008055983105311848317577, −7.24165328333097672634055183361, −5.92263053268278184908595007562, −4.88418676459769637079364624437, −3.98016398475187404182142660499, −3.10199766869741488608791599043, −2.24049083188594880603555223157, −1.24418811897501635113814838757, 0.05562684525111817009255484119, 2.58849700160892322056776833067, 3.404937398361053993035500934198, 3.77382930536445602493921479299, 5.17577454769849665060779947628, 5.815775007358438785984305458515, 7.19273385005840962312673674383, 7.43196939592500137134979320093, 8.54898665210937574874053555638, 9.27610222320517129874921180060, 10.19597532265197630952043330260, 10.84442718201336425870798883025, 12.22479224758669422608028506230, 13.19745059107234305296834521339, 13.76147630184474288818666142317, 14.730377186754728626549416084456, 15.23261338819968056331308880127, 15.84109069359797374291557815404, 16.552128963833344357095904519540, 17.40504594869519064526900447591, 18.56834333082755278816814932610, 19.22623219279859425298578502569, 19.70005342455164826578063110464, 20.99905876727585933431907664104, 21.95222083998886339709466120989

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