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Properties

Label 1-1021-1021.17-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.937 - 0.346i$
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  + (−0.378 + 0.925i)2-s + (0.739 + 0.673i)3-s + (−0.713 − 0.700i)4-s + (−0.823 − 0.567i)5-s + (−0.903 + 0.429i)6-s + (−0.786 − 0.617i)7-s + (0.918 − 0.395i)8-s + (0.0922 + 0.995i)9-s + (0.836 − 0.547i)10-s + (−0.987 − 0.159i)11-s + (−0.0554 − 0.998i)12-s + (0.850 + 0.526i)13-s + (0.869 − 0.494i)14-s + (−0.225 − 0.974i)15-s + (0.0184 + 0.999i)16-s + (−0.521 − 0.853i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.925i)2-s + (0.739 + 0.673i)3-s + (−0.713 − 0.700i)4-s + (−0.823 − 0.567i)5-s + (−0.903 + 0.429i)6-s + (−0.786 − 0.617i)7-s + (0.918 − 0.395i)8-s + (0.0922 + 0.995i)9-s + (0.836 − 0.547i)10-s + (−0.987 − 0.159i)11-s + (−0.0554 − 0.998i)12-s + (0.850 + 0.526i)13-s + (0.869 − 0.494i)14-s + (−0.225 − 0.974i)15-s + (0.0184 + 0.999i)16-s + (−0.521 − 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.937 - 0.346i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.937 - 0.346i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06907826536 + 0.3858483587i\)
\(L(\frac12)\) \(\approx\) \(-0.06907826536 + 0.3858483587i\)
\(L(1)\) \(\approx\) \(0.5904129832 + 0.3593904445i\)
\(L(1)\) \(\approx\) \(0.5904129832 + 0.3593904445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.378 + 0.925i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (-0.823 - 0.567i)T \)
7 \( 1 + (-0.786 - 0.617i)T \)
11 \( 1 + (-0.987 - 0.159i)T \)
13 \( 1 + (0.850 + 0.526i)T \)
17 \( 1 + (-0.521 - 0.853i)T \)
19 \( 1 + (0.908 - 0.417i)T \)
23 \( 1 + (0.856 + 0.515i)T \)
29 \( 1 + (0.141 + 0.989i)T \)
31 \( 1 + (-0.489 - 0.872i)T \)
37 \( 1 + (-0.995 + 0.0984i)T \)
41 \( 1 + (-0.908 + 0.417i)T \)
43 \( 1 + (-0.985 + 0.171i)T \)
47 \( 1 + (-0.00615 + 0.999i)T \)
53 \( 1 + (0.996 - 0.0861i)T \)
59 \( 1 + (0.747 - 0.664i)T \)
61 \( 1 + (-0.862 - 0.505i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (-0.918 + 0.395i)T \)
73 \( 1 + (0.869 + 0.494i)T \)
79 \( 1 + (-0.542 + 0.840i)T \)
83 \( 1 + (-0.952 + 0.303i)T \)
89 \( 1 + (-0.998 - 0.0615i)T \)
97 \( 1 + (0.696 - 0.717i)T \)
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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

−20.91531531310592258782576360820, −20.22093664984120061561959921308, −19.51586690155971351636827696473, −18.90142416982354152424872441104, −18.34523077100995956893341057385, −17.8324089878776255290844802630, −16.43905163402809381150594972128, −15.495823131042134843054206509568, −14.96077268768284236661911802902, −13.63985333571457301725598950669, −13.17570363407705092696159963383, −12.32969660017236371851229397126, −11.784103742396356920154248637902, −10.61455128608070345309149746170, −10.08980082958392059010005391419, −8.765423061360102707977451892947, −8.46664424009249194226554988682, −7.512385799930760387973592400855, −6.72764357096817773773397555715, −5.46860349746696409325540968822, −3.97903686538962553478285230716, −3.222609567489273886773484988397, −2.729760800666692023411075846953, −1.6296100820027851778596682450, −0.18559575578513637615304813220, 1.25672390256436044999299679203, 3.03965226840643133776673107196, 3.812292238191308249441891670031, 4.770507371918479355980342639300, 5.3851392726761421959111057652, 6.88287838624832337836954320823, 7.43898419193128872592924783508, 8.33665976973206875212037060995, 9.042488601936416550579559565231, 9.647866704654781643149445364582, 10.63855160294703531668322847254, 11.39393145257303468065786707708, 13.081224679079515268761962441216, 13.39292427934869604046982926855, 14.21891719026537483472506764651, 15.4346225165520681914730534363, 15.71863553418158809919860389738, 16.34640279603411571419140575245, 16.91419114619140188444487133384, 18.27634681905879096560418112144, 18.886344027149882379245175093495, 19.76598498270559786292800888109, 20.259683261584551426025819938647, 21.080111673674213419738899349193, 22.26138255260664610014809895297

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