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Properties

Label 1-1021-1021.253-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.246 - 0.969i$
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  + (−0.659 + 0.751i)2-s + (−0.602 − 0.798i)3-s + (−0.128 − 0.991i)4-s + (0.997 + 0.0738i)5-s + (0.997 + 0.0738i)6-s + (−0.993 + 0.110i)7-s + (0.830 + 0.557i)8-s + (−0.273 + 0.961i)9-s + (−0.713 + 0.700i)10-s + (0.989 + 0.147i)11-s + (−0.713 + 0.700i)12-s + (0.0922 + 0.995i)13-s + (0.572 − 0.819i)14-s + (−0.542 − 0.840i)15-s + (−0.966 + 0.255i)16-s + (−0.763 − 0.645i)17-s + ⋯
L(s)  = 1  + (−0.659 + 0.751i)2-s + (−0.602 − 0.798i)3-s + (−0.128 − 0.991i)4-s + (0.997 + 0.0738i)5-s + (0.997 + 0.0738i)6-s + (−0.993 + 0.110i)7-s + (0.830 + 0.557i)8-s + (−0.273 + 0.961i)9-s + (−0.713 + 0.700i)10-s + (0.989 + 0.147i)11-s + (−0.713 + 0.700i)12-s + (0.0922 + 0.995i)13-s + (0.572 − 0.819i)14-s + (−0.542 − 0.840i)15-s + (−0.966 + 0.255i)16-s + (−0.763 − 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.246 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4780559432 - 0.3717192296i\)
\(L(\frac12)\) \(\approx\) \(0.4780559432 - 0.3717192296i\)
\(L(1)\) \(\approx\) \(0.6251782084 + 0.01609587511i\)
\(L(1)\) \(\approx\) \(0.6251782084 + 0.01609587511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.659 + 0.751i)T \)
3 \( 1 + (-0.602 - 0.798i)T \)
5 \( 1 + (0.997 + 0.0738i)T \)
7 \( 1 + (-0.993 + 0.110i)T \)
11 \( 1 + (0.989 + 0.147i)T \)
13 \( 1 + (0.0922 + 0.995i)T \)
17 \( 1 + (-0.763 - 0.645i)T \)
19 \( 1 + (-0.273 - 0.961i)T \)
23 \( 1 + (-0.966 - 0.255i)T \)
29 \( 1 + (-0.993 - 0.110i)T \)
31 \( 1 + (-0.343 - 0.938i)T \)
37 \( 1 + (-0.945 + 0.326i)T \)
41 \( 1 + (-0.273 - 0.961i)T \)
43 \( 1 + (-0.201 + 0.979i)T \)
47 \( 1 + (0.572 - 0.819i)T \)
53 \( 1 + (0.631 + 0.775i)T \)
59 \( 1 + (0.956 + 0.291i)T \)
61 \( 1 + (0.572 - 0.819i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.830 + 0.557i)T \)
73 \( 1 + (0.572 + 0.819i)T \)
79 \( 1 + (0.165 - 0.986i)T \)
83 \( 1 + (-0.602 - 0.798i)T \)
89 \( 1 + (-0.982 + 0.183i)T \)
97 \( 1 + (0.739 - 0.673i)T \)
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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

−21.77476396136438120412843337132, −21.00105948724180627735248217372, −20.191667783489616751764316331624, −19.63667734919543013087774052296, −18.534825207313511092985964652, −17.728682375490193332369892999909, −17.17283120905391555758963251810, −16.53654840647450462703029128622, −15.83369970777200206066699435766, −14.70818076646083553639396826668, −13.65737394617523072101747293891, −12.74777078906877249983225341027, −12.22329797428952786295258900779, −11.11427924166084608120470106385, −10.33035166763449398553960549541, −9.93178132128829355982184760041, −9.13010940526912926801189763379, −8.457364256353162187913635082219, −6.92126813915716944871152877528, −6.17174487233522447378594927562, −5.336939337635831390375241237, −3.89587721261068062792218849201, −3.52494631403299396904610180635, −2.21797075933674710151549302062, −1.085259942418767902520184714391, 0.387341756841946158733578034346, 1.74234001863452326060126108701, 2.352856326271171315579277074584, 4.21505276593775164557541991906, 5.36664702782137524979003124883, 6.139696614069311977764692033805, 6.7870752411370508441627438687, 7.11263627155270983578282355738, 8.63609722152725770377005736955, 9.26919860370516989901416751416, 9.935895711884716211859522516065, 10.99277540659371715054512288513, 11.76109136656188296685202866221, 12.909327321932661639815001485591, 13.67183380553512989986967286031, 14.134683317804687453309904269966, 15.31382501160534107310151843666, 16.28317211704845497096905948337, 16.91595664648703487775533474351, 17.389526065352358076497764495696, 18.3148023811153611431573841425, 18.785332938355106573860402375663, 19.61278336579549908180787877339, 20.312620556494478280420589448887, 21.88731587825540174706997776571

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