Properties

Label 1-23e2-529.193-r0-0-0
Degree $1$
Conductor $529$
Sign $0.668 - 0.743i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (−0.993 − 0.111i)2-s + (0.735 − 0.678i)3-s + (0.975 + 0.221i)4-s + (−0.673 + 0.739i)5-s + (−0.806 + 0.591i)6-s + (−0.873 + 0.487i)7-s + (−0.944 − 0.329i)8-s + (0.0806 − 0.996i)9-s + (0.751 − 0.659i)10-s + (0.955 + 0.293i)11-s + (0.867 − 0.498i)12-s + (0.545 − 0.837i)13-s + (0.922 − 0.386i)14-s + (0.00620 + 0.999i)15-s + (0.901 + 0.432i)16-s + (−0.535 − 0.844i)17-s + ⋯
L(s)  = 1  + (−0.993 − 0.111i)2-s + (0.735 − 0.678i)3-s + (0.975 + 0.221i)4-s + (−0.673 + 0.739i)5-s + (−0.806 + 0.591i)6-s + (−0.873 + 0.487i)7-s + (−0.944 − 0.329i)8-s + (0.0806 − 0.996i)9-s + (0.751 − 0.659i)10-s + (0.955 + 0.293i)11-s + (0.867 − 0.498i)12-s + (0.545 − 0.837i)13-s + (0.922 − 0.386i)14-s + (0.00620 + 0.999i)15-s + (0.901 + 0.432i)16-s + (−0.535 − 0.844i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.668 - 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8255962599 - 0.3681845967i\)
\(L(\frac12)\) \(\approx\) \(0.8255962599 - 0.3681845967i\)
\(L(1)\) \(\approx\) \(0.7641579885 - 0.1568147413i\)
\(L(1)\) \(\approx\) \(0.7641579885 - 0.1568147413i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.993 - 0.111i)T \)
3 \( 1 + (0.735 - 0.678i)T \)
5 \( 1 + (-0.673 + 0.739i)T \)
7 \( 1 + (-0.873 + 0.487i)T \)
11 \( 1 + (0.955 + 0.293i)T \)
13 \( 1 + (0.545 - 0.837i)T \)
17 \( 1 + (-0.535 - 0.844i)T \)
19 \( 1 + (-0.896 + 0.443i)T \)
29 \( 1 + (0.645 + 0.763i)T \)
31 \( 1 + (0.626 - 0.779i)T \)
37 \( 1 + (0.767 + 0.640i)T \)
41 \( 1 + (0.346 + 0.938i)T \)
43 \( 1 + (0.890 - 0.454i)T \)
47 \( 1 + (0.682 - 0.730i)T \)
53 \( 1 + (0.751 + 0.659i)T \)
59 \( 1 + (0.0310 - 0.999i)T \)
61 \( 1 + (-0.635 - 0.771i)T \)
67 \( 1 + (0.664 - 0.747i)T \)
71 \( 1 + (0.481 + 0.876i)T \)
73 \( 1 + (0.645 - 0.763i)T \)
79 \( 1 + (0.995 + 0.0991i)T \)
83 \( 1 + (-0.966 + 0.257i)T \)
89 \( 1 + (0.980 - 0.197i)T \)
97 \( 1 + (-0.215 - 0.976i)T \)
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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

−23.85799642206672912071210371004, −22.82546208404118298076400403027, −21.530028493913616761620874903602, −20.936343664337400161207660606026, −19.81848733425485604025027598483, −19.52725858623321670827942208142, −19.029802968536658966434879045099, −17.381314614561739212702921978428, −16.717838538436306652993183039508, −16.03322243309703016865398059180, −15.4588661038133377129070255999, −14.42201121120402725082476108171, −13.362585692726965438522665578086, −12.26871824770677577612734278628, −11.17459460740851301103685603467, −10.43938895814248899182493634023, −9.25697202765935084006130256360, −8.91941031375218370347591485900, −8.087083237292573937520589403602, −6.96596186605525569793173031550, −6.063646047990948725340184717305, −4.30851517239736284684784806055, −3.77084407009639264751689019585, −2.42178034358249007808470076243, −1.046781501008749723543034754190, 0.7574023055547735325156451240, 2.26919717767239352707834590739, 3.01420128701497104311590873166, 3.89163610168061915116544179453, 6.2486184518559993035024238709, 6.63668975166520333784029978519, 7.6216928144477024084376830406, 8.45233930525780241824399711877, 9.2484671994874876066125642134, 10.13577834804001368901527950980, 11.25207176546529805017657668471, 12.11959779471605337167443717785, 12.80248836830924177427232167474, 14.072883854851269537036800058784, 15.19487338097611511886288634553, 15.50472524073295561069816931518, 16.67940674017113116210194824048, 17.81902429604787793320235990766, 18.52458391494438057316829760353, 19.05804683127704018214630390489, 19.913023300183056765981557821305, 20.25750279615604764145185182743, 21.58552691230998480112270994211, 22.62638343758204220558544643513, 23.40423892475449173149508396182

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