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Properties

Label 1-269-269.21-r0-0-0
Degree $1$
Conductor $269$
Sign $0.454 + 0.890i$
Analytic cond. $1.24923$
Root an. cond. $1.24923$
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  + (−0.990 + 0.140i)2-s + (0.930 + 0.366i)3-s + (0.960 − 0.277i)4-s + (−0.553 + 0.833i)5-s + (−0.972 − 0.232i)6-s + (0.892 + 0.451i)7-s + (−0.912 + 0.409i)8-s + (0.731 + 0.681i)9-s + (0.430 − 0.902i)10-s + (0.591 − 0.806i)11-s + (0.995 + 0.0936i)12-s + (0.430 − 0.902i)13-s + (−0.946 − 0.322i)14-s + (−0.819 + 0.572i)15-s + (0.845 − 0.533i)16-s + (0.591 + 0.806i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.140i)2-s + (0.930 + 0.366i)3-s + (0.960 − 0.277i)4-s + (−0.553 + 0.833i)5-s + (−0.972 − 0.232i)6-s + (0.892 + 0.451i)7-s + (−0.912 + 0.409i)8-s + (0.731 + 0.681i)9-s + (0.430 − 0.902i)10-s + (0.591 − 0.806i)11-s + (0.995 + 0.0936i)12-s + (0.430 − 0.902i)13-s + (−0.946 − 0.322i)14-s + (−0.819 + 0.572i)15-s + (0.845 − 0.533i)16-s + (0.591 + 0.806i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(269\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(1.24923\)
Root analytic conductor: \(1.24923\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 269,\ (0:\ ),\ 0.454 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9892321037 + 0.6059675927i\)
\(L(\frac12)\) \(\approx\) \(0.9892321037 + 0.6059675927i\)
\(L(1)\) \(\approx\) \(0.9462200018 + 0.3295537280i\)
\(L(1)\) \(\approx\) \(0.9462200018 + 0.3295537280i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 \)
good2 \( 1 + (-0.990 + 0.140i)T \)
3 \( 1 + (0.930 + 0.366i)T \)
5 \( 1 + (-0.553 + 0.833i)T \)
7 \( 1 + (0.892 + 0.451i)T \)
11 \( 1 + (0.591 - 0.806i)T \)
13 \( 1 + (0.430 - 0.902i)T \)
17 \( 1 + (0.591 + 0.806i)T \)
19 \( 1 + (-0.998 - 0.0468i)T \)
23 \( 1 + (0.930 + 0.366i)T \)
29 \( 1 + (-0.388 + 0.921i)T \)
31 \( 1 + (-0.300 - 0.953i)T \)
37 \( 1 + (-0.0234 - 0.999i)T \)
41 \( 1 + (-0.990 - 0.140i)T \)
43 \( 1 + (-0.388 + 0.921i)T \)
47 \( 1 + (-0.209 + 0.977i)T \)
53 \( 1 + (-0.762 - 0.646i)T \)
59 \( 1 + (0.960 - 0.277i)T \)
61 \( 1 + (-0.116 - 0.993i)T \)
67 \( 1 + (0.960 + 0.277i)T \)
71 \( 1 + (-0.946 + 0.322i)T \)
73 \( 1 + (0.792 + 0.610i)T \)
79 \( 1 + (-0.472 + 0.881i)T \)
83 \( 1 + (0.0702 + 0.997i)T \)
89 \( 1 + (-0.869 + 0.493i)T \)
97 \( 1 + (-0.116 + 0.993i)T \)
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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

−25.46897551558829263331919549202, −24.99575048201082266395211919782, −24.019008960355566983914866251849, −23.34618845512948446610204884261, −21.3144834561305537939792091800, −20.684619721437445418925878871563, −20.142197818819847125883803769243, −19.18980707045361585990833908409, −18.47640564559514012337192183337, −17.27908839151038672545529001530, −16.61297686037095321750993216324, −15.37174465642589452685580979650, −14.62251259396561158987734564176, −13.3705057856622844443990443802, −12.18073723795064875274790944289, −11.54104387504529128619582683862, −10.140501465455683291271932076249, −9.01866508342846466916637966248, −8.49466621848926217864771613282, −7.48648427388634350291064857294, −6.758392983088001103155168172079, −4.67615569475791015919887921767, −3.61688245313658390425437676862, −1.98425937599347077826557481916, −1.172274370892433656309421714824, 1.558803018738294158352508597286, 2.83887977017650062026399092335, 3.7750827962854527502672966411, 5.58040471650606931961934803060, 6.881694700979648280791131729257, 8.10856755985186913437229514620, 8.37141191274237009908790522295, 9.58026417517832297031028540040, 10.865588397289688713385675232526, 11.14090110501110939801277659886, 12.70133703782209717661737654589, 14.35910462333028297456143918714, 14.8834054427804417004440534418, 15.54962352516120783263072279381, 16.67631772777451752712572290579, 17.80419575432729975839964820177, 18.85457417893522653252545120494, 19.23182699662300574910752085788, 20.26853863816566280823107919711, 21.19438557076619373252385162262, 21.95643735475981930336145227040, 23.48702579257514219917831229985, 24.41192960620840526359787172126, 25.333299273031115166221911631836, 25.92950392728254178626364435549

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