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Properties

Label 1-1021-1021.1001-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.998 + 0.0545i$
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  + (−0.966 − 0.255i)2-s + (−0.850 + 0.526i)3-s + (0.869 + 0.494i)4-s + (−0.225 + 0.974i)5-s + (0.956 − 0.291i)6-s + (0.903 + 0.429i)7-s + (−0.713 − 0.700i)8-s + (0.445 − 0.895i)9-s + (0.467 − 0.883i)10-s + (0.0677 + 0.997i)11-s + (−0.999 + 0.0369i)12-s + (0.932 + 0.361i)13-s + (−0.763 − 0.645i)14-s + (−0.320 − 0.947i)15-s + (0.510 + 0.859i)16-s + (0.189 + 0.981i)17-s + ⋯
L(s)  = 1  + (−0.966 − 0.255i)2-s + (−0.850 + 0.526i)3-s + (0.869 + 0.494i)4-s + (−0.225 + 0.974i)5-s + (0.956 − 0.291i)6-s + (0.903 + 0.429i)7-s + (−0.713 − 0.700i)8-s + (0.445 − 0.895i)9-s + (0.467 − 0.883i)10-s + (0.0677 + 0.997i)11-s + (−0.999 + 0.0369i)12-s + (0.932 + 0.361i)13-s + (−0.763 − 0.645i)14-s + (−0.320 − 0.947i)15-s + (0.510 + 0.859i)16-s + (0.189 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.998 + 0.0545i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.998 + 0.0545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01343769729 + 0.4926550384i\)
\(L(\frac12)\) \(\approx\) \(0.01343769729 + 0.4926550384i\)
\(L(1)\) \(\approx\) \(0.4718786040 + 0.2578000870i\)
\(L(1)\) \(\approx\) \(0.4718786040 + 0.2578000870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.966 - 0.255i)T \)
3 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (-0.225 + 0.974i)T \)
7 \( 1 + (0.903 + 0.429i)T \)
11 \( 1 + (0.0677 + 0.997i)T \)
13 \( 1 + (0.932 + 0.361i)T \)
17 \( 1 + (0.189 + 0.981i)T \)
19 \( 1 + (-0.998 - 0.0615i)T \)
23 \( 1 + (-0.999 - 0.0123i)T \)
29 \( 1 + (-0.823 - 0.567i)T \)
31 \( 1 + (0.771 - 0.636i)T \)
37 \( 1 + (-0.960 - 0.279i)T \)
41 \( 1 + (-0.998 - 0.0615i)T \)
43 \( 1 + (0.285 + 0.958i)T \)
47 \( 1 + (-0.177 + 0.984i)T \)
53 \( 1 + (0.801 + 0.597i)T \)
59 \( 1 + (0.612 + 0.790i)T \)
61 \( 1 + (0.941 - 0.338i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (-0.713 - 0.700i)T \)
73 \( 1 + (-0.763 + 0.645i)T \)
79 \( 1 + (0.786 - 0.617i)T \)
83 \( 1 + (0.881 + 0.473i)T \)
89 \( 1 + (0.213 - 0.976i)T \)
97 \( 1 + (0.332 - 0.943i)T \)
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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

−20.883583787528254106396576806, −20.478567983411704866399607478041, −19.43286297499129228746345989781, −18.74544258755835571389409860916, −17.94048496352975699324053713669, −17.38950941644653637005188513259, −16.4927693167715955107379400820, −16.24480190471829891659106594899, −15.256806564065418915511592536576, −13.96484419258006055088851156951, −13.32634208986434112754772305744, −12.06852316752070276891966704560, −11.60465326709583489526816488537, −10.80492084292470915888970026573, −10.1301680476941511323039943060, −8.62755405345648119806257450386, −8.41700192005630120078522477300, −7.481511320566687824156657574994, −6.58075601777010283468707093173, −5.59300995104482055703209016979, −5.05300562201042819737734917437, −3.727685519146116650212132443238, −1.997918921449547213291118748121, −1.1986628201452071388589257639, −0.357070802477846984467812484974, 1.509417973836582444811160046020, 2.28884174251567688006947268057, 3.72905701779301047113760465494, 4.3568823525504780895566310474, 5.91149839699049474681211974164, 6.40228967290566384052971825231, 7.43226823741115721002984995186, 8.26274643355378041535785582476, 9.21100569209623901593294119716, 10.29614581026064637947222765087, 10.57579117959078414920627081804, 11.58724247052261518972171072494, 11.858823976510910320681181658419, 12.953718792524557465895521292962, 14.516632761790301353961622935359, 15.1982068278275193501321247074, 15.653123422425974596093511619238, 16.73714275966283107429295798258, 17.54139206916781337548230102859, 17.9383287961929189121668116751, 18.71185453873258928050434234482, 19.39584194712973597049656182603, 20.661154230792112101217879079635, 21.073812395795352431987629754418, 21.86771205009771887854219099358

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