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Properties

Label 1-1021-1021.2-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.885 + 0.465i$
Analytic cond. $109.721$
Root an. cond. $109.721$
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  + (−0.219 − 0.975i)2-s + (0.602 + 0.798i)3-s + (−0.903 + 0.429i)4-s + (−0.763 − 0.645i)5-s + (0.645 − 0.763i)6-s + (−0.494 + 0.869i)7-s + (0.617 + 0.786i)8-s + (−0.273 + 0.961i)9-s + (−0.462 + 0.886i)10-s + (0.165 + 0.986i)11-s + (−0.886 − 0.462i)12-s + (0.995 − 0.0922i)13-s + (0.956 + 0.291i)14-s + (0.0554 − 0.998i)15-s + (0.631 − 0.775i)16-s + (−0.378 + 0.925i)17-s + ⋯
L(s)  = 1  + (−0.219 − 0.975i)2-s + (0.602 + 0.798i)3-s + (−0.903 + 0.429i)4-s + (−0.763 − 0.645i)5-s + (0.645 − 0.763i)6-s + (−0.494 + 0.869i)7-s + (0.617 + 0.786i)8-s + (−0.273 + 0.961i)9-s + (−0.462 + 0.886i)10-s + (0.165 + 0.986i)11-s + (−0.886 − 0.462i)12-s + (0.995 − 0.0922i)13-s + (0.956 + 0.291i)14-s + (0.0554 − 0.998i)15-s + (0.631 − 0.775i)16-s + (−0.378 + 0.925i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.885 + 0.465i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.885 + 0.465i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2612037072 + 1.057702407i\)
\(L(\frac12)\) \(\approx\) \(0.2612037072 + 1.057702407i\)
\(L(1)\) \(\approx\) \(0.8357604741 + 0.1471954124i\)
\(L(1)\) \(\approx\) \(0.8357604741 + 0.1471954124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.219 - 0.975i)T \)
3 \( 1 + (0.602 + 0.798i)T \)
5 \( 1 + (-0.763 - 0.645i)T \)
7 \( 1 + (-0.494 + 0.869i)T \)
11 \( 1 + (0.165 + 0.986i)T \)
13 \( 1 + (0.995 - 0.0922i)T \)
17 \( 1 + (-0.378 + 0.925i)T \)
19 \( 1 + (-0.961 + 0.273i)T \)
23 \( 1 + (0.631 + 0.775i)T \)
29 \( 1 + (-0.869 + 0.494i)T \)
31 \( 1 + (-0.557 + 0.830i)T \)
37 \( 1 + (0.999 - 0.0184i)T \)
41 \( 1 + (0.273 + 0.961i)T \)
43 \( 1 + (0.911 - 0.412i)T \)
47 \( 1 + (0.956 + 0.291i)T \)
53 \( 1 + (0.840 + 0.542i)T \)
59 \( 1 + (0.326 + 0.945i)T \)
61 \( 1 + (-0.956 - 0.291i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.786 - 0.617i)T \)
73 \( 1 + (0.956 - 0.291i)T \)
79 \( 1 + (0.713 - 0.700i)T \)
83 \( 1 + (0.602 + 0.798i)T \)
89 \( 1 + (-0.982 + 0.183i)T \)
97 \( 1 + (-0.673 - 0.739i)T \)
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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

−20.90506194834276593943744894771, −19.99854551709413117042173850307, −19.21156724297128810967818011888, −18.755335504099557768130349204556, −18.15267947603137734667109873984, −17.05414178888457190262626247814, −16.36176387259031204163575550240, −15.535804522786297579584986605515, −14.76281056857891296450546569234, −13.98302973384283206967651418416, −13.4205575001880955858011574459, −12.7195720004241163087476111541, −11.295494392454164195293179820131, −10.735890657423531495537422866781, −9.41215707881241165009103180759, −8.70075502079135224268222080102, −7.91641192238162111717494682605, −7.17257336090552615437957946654, −6.57983302658445119038453446739, −5.88191493106650030238943532347, −4.1720951959211929795426158387, −3.70057219593840059932450168753, −2.547225952193756659592276949858, −0.82250908699721367877647061097, −0.31172426608860589521360853473, 1.39398636461903680283825063845, 2.330739451224449651655304753507, 3.44922502483499776395753649521, 4.01382819543371800503689021206, 4.81954640297226996266408072967, 5.83945222533938213186919027725, 7.49082845976957197364208170951, 8.3891939482356829305888043411, 9.01863172979627880196827150561, 9.44929567633840333459871921169, 10.64769888721865171790771358467, 11.14996274814036689974973964850, 12.31502854276464695862027225800, 12.77910821465042714784742403958, 13.57265597694264501401676632553, 14.986579667687116438872705819771, 15.19489605633260397194478669323, 16.30917936821073536496487445029, 16.94251836726036238161969088620, 18.054321964139651944617768364434, 18.94037921677003014362071130928, 19.66724825657654681644306314763, 20.03824048814386821131972991721, 21.001378717264808664537989439230, 21.40871393245864744190308333091

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