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Properties

Label 1-1021-1021.335-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.875 - 0.483i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.875 - 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1480844830 - 0.03818692090i\)
\(L(\frac12)\) \(\approx\) \(0.1480844830 - 0.03818692090i\)
\(L(1)\) \(\approx\) \(0.5791893963 + 0.5986278037i\)
\(L(1)\) \(\approx\) \(0.5791893963 + 0.5986278037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.237 + 0.971i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (-0.478 + 0.878i)T \)
7 \( 1 + (-0.999 - 0.0369i)T \)
11 \( 1 + (-0.542 - 0.840i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
17 \( 1 + (0.687 + 0.726i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (0.572 + 0.819i)T \)
29 \( 1 + (-0.999 + 0.0369i)T \)
31 \( 1 + (-0.918 + 0.395i)T \)
37 \( 1 + (-0.993 - 0.110i)T \)
41 \( 1 + (0.0922 - 0.995i)T \)
43 \( 1 + (0.830 - 0.557i)T \)
47 \( 1 + (-0.201 - 0.979i)T \)
53 \( 1 + (0.956 - 0.291i)T \)
59 \( 1 + (-0.412 + 0.911i)T \)
61 \( 1 + (-0.201 - 0.979i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.659 - 0.751i)T \)
73 \( 1 + (-0.201 + 0.979i)T \)
79 \( 1 + (-0.0554 - 0.998i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.273 - 0.961i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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