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Properties

Label 1-1021-1021.5-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.994 - 0.105i$
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  + (−0.763 − 0.645i)2-s + (0.739 + 0.673i)3-s + (0.165 + 0.986i)4-s + (0.923 + 0.384i)5-s + (−0.128 − 0.991i)6-s + (0.830 − 0.557i)7-s + (0.510 − 0.859i)8-s + (0.0922 + 0.995i)9-s + (−0.456 − 0.889i)10-s + (0.261 − 0.965i)11-s + (−0.542 + 0.840i)12-s + (−0.850 − 0.526i)13-s + (−0.993 − 0.110i)14-s + (0.423 + 0.905i)15-s + (−0.945 + 0.326i)16-s + (−0.823 − 0.567i)17-s + ⋯
L(s)  = 1  + (−0.763 − 0.645i)2-s + (0.739 + 0.673i)3-s + (0.165 + 0.986i)4-s + (0.923 + 0.384i)5-s + (−0.128 − 0.991i)6-s + (0.830 − 0.557i)7-s + (0.510 − 0.859i)8-s + (0.0922 + 0.995i)9-s + (−0.456 − 0.889i)10-s + (0.261 − 0.965i)11-s + (−0.542 + 0.840i)12-s + (−0.850 − 0.526i)13-s + (−0.993 − 0.110i)14-s + (0.423 + 0.905i)15-s + (−0.945 + 0.326i)16-s + (−0.823 − 0.567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.994 - 0.105i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.994 - 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.777248297 - 0.09406949414i\)
\(L(\frac12)\) \(\approx\) \(1.777248297 - 0.09406949414i\)
\(L(1)\) \(\approx\) \(1.219294905 - 0.06518882548i\)
\(L(1)\) \(\approx\) \(1.219294905 - 0.06518882548i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.763 - 0.645i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (0.923 + 0.384i)T \)
7 \( 1 + (0.830 - 0.557i)T \)
11 \( 1 + (0.261 - 0.965i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
17 \( 1 + (-0.823 - 0.567i)T \)
19 \( 1 + (0.816 + 0.577i)T \)
23 \( 1 + (0.189 + 0.981i)T \)
29 \( 1 + (0.0677 - 0.997i)T \)
31 \( 1 + (-0.297 - 0.954i)T \)
37 \( 1 + (0.949 + 0.314i)T \)
41 \( 1 + (0.816 + 0.577i)T \)
43 \( 1 + (0.531 - 0.846i)T \)
47 \( 1 + (0.401 + 0.916i)T \)
53 \( 1 + (-0.875 + 0.483i)T \)
59 \( 1 + (-0.00615 + 0.999i)T \)
61 \( 1 + (0.592 - 0.805i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (0.510 - 0.859i)T \)
73 \( 1 + (-0.993 + 0.110i)T \)
79 \( 1 + (0.631 + 0.775i)T \)
83 \( 1 + (0.213 - 0.976i)T \)
89 \( 1 + (0.552 - 0.833i)T \)
97 \( 1 + (0.969 + 0.243i)T \)
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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

−21.49639716770063061649604442669, −20.49136829674841985663409050367, −20.01626540708543310831949134676, −19.2061281916427152399467639327, −18.131885843225236492381469180921, −17.861345719395343410431071872792, −17.2528465068652093571654848648, −16.19987547236831482140611153288, −15.14671676690151771182370917289, −14.480130785265720021099199436394, −14.11994165062511701156212357874, −12.92246963797245100367483977653, −12.233721328496605885234831670653, −11.10585127331961054536146525588, −10.00408227512315674445548497947, −9.10219017367628291143423800932, −8.897692721039643023126925320448, −7.8279439530036779075345286000, −7.00542017366538233882917745806, −6.35911080619943851014868076700, −5.2002373729605474524001059248, −4.517578326886379707219517318340, −2.48570530392115546177051943164, −2.00354937409594360971355056113, −1.143134283570185081247718202599, 1.083600880253439714058258637726, 2.20855984114904891776050824579, 2.874812351477253598780996578314, 3.84482737943838877507664775458, 4.82144543450753938592960896796, 5.9504925582189393083058902182, 7.44970864572521498740011749406, 7.80575379572709727852040837187, 8.95557540900959250915762341817, 9.56407230906853920432136790371, 10.21037710097683451266705649716, 11.06288775908843412926518820958, 11.56300892107915137233662869621, 13.07666359039122339003902889246, 13.70612848524716967231988097797, 14.32489246962789906936839128620, 15.31155829105713986262953594323, 16.28812025715994612979393282737, 17.12076855357464097069013476257, 17.62948911244172756965793114783, 18.56843823659915521169341126199, 19.31263307097856391363350514057, 20.21561117781844867663124630120, 20.66208078758788168167869943426, 21.44443214103782046037331484337

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