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Properties

Label 1-1021-1021.192-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.943 - 0.330i$
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  + (−0.918 + 0.395i)2-s + (−0.602 + 0.798i)3-s + (0.687 − 0.726i)4-s + (0.237 − 0.971i)5-s + (0.237 − 0.971i)6-s + (−0.412 + 0.911i)7-s + (−0.343 + 0.938i)8-s + (−0.273 − 0.961i)9-s + (0.165 + 0.986i)10-s + (−0.886 − 0.462i)11-s + (0.165 + 0.986i)12-s + (0.0922 − 0.995i)13-s + (0.0184 − 0.999i)14-s + (0.631 + 0.775i)15-s + (−0.0554 − 0.998i)16-s + (0.997 − 0.0738i)17-s + ⋯
L(s)  = 1  + (−0.918 + 0.395i)2-s + (−0.602 + 0.798i)3-s + (0.687 − 0.726i)4-s + (0.237 − 0.971i)5-s + (0.237 − 0.971i)6-s + (−0.412 + 0.911i)7-s + (−0.343 + 0.938i)8-s + (−0.273 − 0.961i)9-s + (0.165 + 0.986i)10-s + (−0.886 − 0.462i)11-s + (0.165 + 0.986i)12-s + (0.0922 − 0.995i)13-s + (0.0184 − 0.999i)14-s + (0.631 + 0.775i)15-s + (−0.0554 − 0.998i)16-s + (0.997 − 0.0738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.943 - 0.330i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.943 - 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009203760295 + 0.05415138329i\)
\(L(\frac12)\) \(\approx\) \(0.009203760295 + 0.05415138329i\)
\(L(1)\) \(\approx\) \(0.4509707832 + 0.1127432763i\)
\(L(1)\) \(\approx\) \(0.4509707832 + 0.1127432763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.918 + 0.395i)T \)
3 \( 1 + (-0.602 + 0.798i)T \)
5 \( 1 + (0.237 - 0.971i)T \)
7 \( 1 + (-0.412 + 0.911i)T \)
11 \( 1 + (-0.886 - 0.462i)T \)
13 \( 1 + (0.0922 - 0.995i)T \)
17 \( 1 + (0.997 - 0.0738i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (-0.0554 + 0.998i)T \)
29 \( 1 + (-0.412 - 0.911i)T \)
31 \( 1 + (-0.999 - 0.0369i)T \)
37 \( 1 + (0.956 - 0.291i)T \)
41 \( 1 + (-0.273 + 0.961i)T \)
43 \( 1 + (0.869 - 0.494i)T \)
47 \( 1 + (0.0184 - 0.999i)T \)
53 \( 1 + (-0.966 + 0.255i)T \)
59 \( 1 + (0.572 + 0.819i)T \)
61 \( 1 + (0.0184 - 0.999i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.343 + 0.938i)T \)
73 \( 1 + (0.0184 + 0.999i)T \)
79 \( 1 + (0.989 + 0.147i)T \)
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

−21.04299079294737545271630218312, −20.23091023755536291760510926504, −19.189342404982215483078535363734, −18.912227297031278810552908841153, −18.03600984399728309020762826492, −17.56315367068084140209518756924, −16.57486457998558868564868230545, −16.18693944666640784042644742479, −14.83340979964491234628471243385, −13.897759927915789848670132212157, −12.993107937401720356268468790214, −12.392326439819757159196354794235, −11.23741948302989625795600763679, −10.808107896390366873348345279775, −10.13456443454110791462014195279, −9.21043454365700554006197268085, −7.90674657774911520815076088983, −7.24552702673666057554469182316, −6.789165864410255129778654179277, −5.90940741696629197355789384746, −4.45528518064753628604229874381, −3.17523838603940388241998891750, −2.34611678067261290520972073598, −1.40034427708670922743260598884, −0.03756603811989286774025066023, 1.18462796870828022453623775190, 2.55818415107487072506206887839, 3.69284036197214074893539646775, 5.289649290575832397423529156946, 5.54902002961843698991866386776, 6.13945484549422491540002847431, 7.75433242344030340027934741253, 8.32307783928668663583535987659, 9.33274275664720722570499794930, 9.79232556269740269952073308328, 10.5758580129108864688528388218, 11.546898852510748911571577088736, 12.32374818708280871424390851857, 13.15454054198060672275314922656, 14.54675119567196394228799783635, 15.38713407364692762323282985670, 15.91600007152140047282441602074, 16.55323209221794317262407462550, 17.16219353588654033340768664982, 18.09709547687511167109552441636, 18.63548871805870242923812395692, 19.69121129751676397807811973031, 20.58329912144611865557860690311, 21.1094374901012825464080798377, 21.821591696078821542365723328465

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