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Properties

Label 1-1021-1021.104-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.819 - 0.573i$
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  + (0.687 + 0.726i)2-s + (−0.273 + 0.961i)3-s + (−0.0554 + 0.998i)4-s + (−0.886 + 0.462i)5-s + (−0.886 + 0.462i)6-s + (−0.659 + 0.751i)7-s + (−0.763 + 0.645i)8-s + (−0.850 − 0.526i)9-s + (−0.945 − 0.326i)10-s + (0.572 − 0.819i)11-s + (−0.945 − 0.326i)12-s + (−0.982 + 0.183i)13-s + (−0.999 + 0.0369i)14-s + (−0.201 − 0.979i)15-s + (−0.993 − 0.110i)16-s + (0.989 + 0.147i)17-s + ⋯
L(s)  = 1  + (0.687 + 0.726i)2-s + (−0.273 + 0.961i)3-s + (−0.0554 + 0.998i)4-s + (−0.886 + 0.462i)5-s + (−0.886 + 0.462i)6-s + (−0.659 + 0.751i)7-s + (−0.763 + 0.645i)8-s + (−0.850 − 0.526i)9-s + (−0.945 − 0.326i)10-s + (0.572 − 0.819i)11-s + (−0.945 − 0.326i)12-s + (−0.982 + 0.183i)13-s + (−0.999 + 0.0369i)14-s + (−0.201 − 0.979i)15-s + (−0.993 − 0.110i)16-s + (0.989 + 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.819 - 0.573i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.819 - 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1757121240 + 0.05535409646i\)
\(L(\frac12)\) \(\approx\) \(-0.1757121240 + 0.05535409646i\)
\(L(1)\) \(\approx\) \(0.4646255832 + 0.6642065456i\)
\(L(1)\) \(\approx\) \(0.4646255832 + 0.6642065456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.687 + 0.726i)T \)
3 \( 1 + (-0.273 + 0.961i)T \)
5 \( 1 + (-0.886 + 0.462i)T \)
7 \( 1 + (-0.659 + 0.751i)T \)
11 \( 1 + (0.572 - 0.819i)T \)
13 \( 1 + (-0.982 + 0.183i)T \)
17 \( 1 + (0.989 + 0.147i)T \)
19 \( 1 + (-0.850 + 0.526i)T \)
23 \( 1 + (-0.993 + 0.110i)T \)
29 \( 1 + (-0.659 - 0.751i)T \)
31 \( 1 + (0.997 - 0.0738i)T \)
37 \( 1 + (0.830 + 0.557i)T \)
41 \( 1 + (-0.850 + 0.526i)T \)
43 \( 1 + (0.510 + 0.859i)T \)
47 \( 1 + (-0.999 + 0.0369i)T \)
53 \( 1 + (0.869 + 0.494i)T \)
59 \( 1 + (-0.343 - 0.938i)T \)
61 \( 1 + (-0.999 + 0.0369i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.763 + 0.645i)T \)
73 \( 1 + (-0.999 - 0.0369i)T \)
79 \( 1 + (0.956 - 0.291i)T \)
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

−20.589789759317740914383361088721, −19.999799137038112386579944221491, −19.48520823991923941542243083474, −18.991565605479448473591235122296, −17.88418676376270847831897513957, −16.98072525460987927421074750899, −16.28546981631025863385423569041, −15.09320062566603922295496812023, −14.44618550575002515434392551541, −13.50374356841181739324606518696, −12.73187557264863629381161370394, −12.2171644786043620475995320113, −11.71249455040415589822754600410, −10.62455994638038631486448785021, −9.85845540526403727803632828448, −8.79988196398454439619317415665, −7.54060716366090887202347102455, −7.02858228742366886980552200254, −6.057315996068188585856566051184, −4.96865286875585446550542106139, −4.20220546165416523216899732719, −3.262038974597697037181032587162, −2.181270087280726201827282429889, −1.09691830029876224471335161887, −0.07035921807492099665082955552, 2.623533541358288857697058434639, 3.37981566223778956304164145992, 4.06118813400835462049347350239, 4.93062467612829912559787537262, 6.1228756103944808849632490192, 6.31170996343730556736344629673, 7.75893334945341281551234973271, 8.38700249911638183076240474617, 9.407101213630924160335287812346, 10.23063655731708780701754073513, 11.56357999816036139368694443386, 11.84130064930895100974712292149, 12.660863114574901742533150294518, 13.959611719640820328480805826475, 14.77574149224528993678433999647, 15.11095774127634624947569832137, 16.0078249047163184491173009071, 16.573234247029267556228013696229, 17.1359238893514331431132214104, 18.408673464300987559114491435397, 19.22627459006599823743771282387, 20.01153549332713759336125771683, 21.216117793055535229954113257807, 21.71074879141810705796698283756, 22.379916448233920055997933257875

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