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Properties

Label 1-1021-1021.262-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.171 + 0.985i$
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  + (0.997 + 0.0738i)2-s + (0.739 + 0.673i)3-s + (0.989 + 0.147i)4-s + (0.687 + 0.726i)5-s + (0.687 + 0.726i)6-s + (−0.343 + 0.938i)7-s + (0.975 + 0.219i)8-s + (0.0922 + 0.995i)9-s + (0.631 + 0.775i)10-s + (−0.0554 + 0.998i)11-s + (0.631 + 0.775i)12-s + (−0.850 − 0.526i)13-s + (−0.412 + 0.911i)14-s + (0.0184 + 0.999i)15-s + (0.956 + 0.291i)16-s + (−0.128 − 0.991i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0738i)2-s + (0.739 + 0.673i)3-s + (0.989 + 0.147i)4-s + (0.687 + 0.726i)5-s + (0.687 + 0.726i)6-s + (−0.343 + 0.938i)7-s + (0.975 + 0.219i)8-s + (0.0922 + 0.995i)9-s + (0.631 + 0.775i)10-s + (−0.0554 + 0.998i)11-s + (0.631 + 0.775i)12-s + (−0.850 − 0.526i)13-s + (−0.412 + 0.911i)14-s + (0.0184 + 0.999i)15-s + (0.956 + 0.291i)16-s + (−0.128 − 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.171 + 0.985i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.171 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.621437924 + 3.115634321i\)
\(L(\frac12)\) \(\approx\) \(2.621437924 + 3.115634321i\)
\(L(1)\) \(\approx\) \(2.236993419 + 1.292593199i\)
\(L(1)\) \(\approx\) \(2.236993419 + 1.292593199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.997 + 0.0738i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (0.687 + 0.726i)T \)
7 \( 1 + (-0.343 + 0.938i)T \)
11 \( 1 + (-0.0554 + 0.998i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
17 \( 1 + (-0.128 - 0.991i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (0.956 - 0.291i)T \)
29 \( 1 + (-0.343 - 0.938i)T \)
31 \( 1 + (-0.659 - 0.751i)T \)
37 \( 1 + (0.869 - 0.494i)T \)
41 \( 1 + (0.0922 - 0.995i)T \)
43 \( 1 + (0.786 + 0.617i)T \)
47 \( 1 + (-0.412 + 0.911i)T \)
53 \( 1 + (-0.945 - 0.326i)T \)
59 \( 1 + (-0.993 - 0.110i)T \)
61 \( 1 + (-0.412 + 0.911i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.975 + 0.219i)T \)
73 \( 1 + (-0.412 - 0.911i)T \)
79 \( 1 + (-0.966 - 0.255i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.273 - 0.961i)T \)
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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

−21.47334810275487263516110331285, −20.573060406257683106082739083288, −19.96866853119006887406423919225, −19.37197450570103695910431985296, −18.5113130562651024950688336539, −17.0860375846530731778884888080, −16.7920305796143856562171930815, −15.835352057963461545125837632534, −14.5185571926807459465428993357, −14.28617634070200580828221786362, −13.354758963265486234692921325541, −12.90798924728927112151662423773, −12.26591756468235009638963988318, −11.12041831216715598263774386504, −10.17730756425202656263948671644, −9.28912805873601312681320830604, −8.270111564906876952595698975199, −7.38692314289749312932083589112, −6.54001135805792539673516976749, −5.8204592835621540205226538803, −4.73782149880968929721157639404, −3.7356011946365694750188311251, −3.01728340330815610582659301121, −1.81037503631326286684496301320, −1.14996641113815621173876268748, 2.15282853770105767457920229855, 2.55030325723770052499187095078, 3.21055238442437989456078794938, 4.54335999093310173869810583361, 5.1435235426919664197103062162, 6.05429858394649069365474788764, 7.14521342408461182959919417486, 7.71039629519832968131791352902, 9.340490073914278659298376586608, 9.56551848143680584826139849889, 10.69815099496123435217365631116, 11.41016328909142632405632346222, 12.594760498233683223644967199395, 13.170494872652234714309517557129, 14.06508680886391019054452017029, 14.87785990255593106281954386111, 15.1853943212963356472426104984, 15.89853859614511222029568536473, 16.97699108938926218799876063577, 17.86437510686444058332575737649, 18.93954056961905476084352722866, 19.68068042939143147381260901157, 20.53593612976536134642195247905, 21.145756711770289910855606530522, 21.92086800735454262862076106971

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