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Properties

Label 1-1021-1021.108-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: $\chi_{1021} (108, \cdot )$
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

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

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