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Properties

Label 1-1021-1021.255-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  + (0.903 + 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)  = 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

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

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: $\chi_{1021} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.875 - 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3248320446 - 1.259866412i\)
\(L(\frac12)\) \(\approx\) \(0.3248320446 - 1.259866412i\)
\(L(1)\) \(\approx\) \(1.203231001 - 0.4812203880i\)
\(L(1)\) \(\approx\) \(1.203231001 - 0.4812203880i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.903 + 0.429i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (0.165 - 0.986i)T \)
7 \( 1 + (0.510 - 0.859i)T \)
11 \( 1 + (-0.945 - 0.326i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (-0.713 + 0.700i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (-0.201 - 0.979i)T \)
29 \( 1 + (0.510 + 0.859i)T \)
31 \( 1 + (0.378 - 0.925i)T \)
37 \( 1 + (-0.999 - 0.0369i)T \)
41 \( 1 + (-0.850 - 0.526i)T \)
43 \( 1 + (-0.659 - 0.751i)T \)
47 \( 1 + (0.830 - 0.557i)T \)
53 \( 1 + (-0.412 + 0.911i)T \)
59 \( 1 + (0.786 + 0.617i)T \)
61 \( 1 + (0.830 - 0.557i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.237 + 0.971i)T \)
73 \( 1 + (0.830 + 0.557i)T \)
79 \( 1 + (0.0184 + 0.999i)T \)
83 \( 1 + (-0.273 - 0.961i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (0.0922 + 0.995i)T \)
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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

−21.93702152373165729385963467177, −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