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Properties

Label 1-571-571.228-r0-0-0
Degree $1$
Conductor $571$
Sign $0.488 + 0.872i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (0.789 + 0.614i)2-s + (−0.213 − 0.976i)3-s + (0.245 + 0.969i)4-s + (0.922 + 0.386i)5-s + (0.431 − 0.901i)6-s + (−0.574 + 0.818i)7-s + (−0.401 + 0.915i)8-s + (−0.909 + 0.416i)9-s + (0.490 + 0.871i)10-s + (0.490 − 0.871i)11-s + (0.894 − 0.446i)12-s + (0.965 − 0.261i)13-s + (−0.956 + 0.293i)14-s + (0.180 − 0.983i)15-s + (−0.879 + 0.475i)16-s + (−0.724 − 0.689i)17-s + ⋯
L(s)  = 1  + (0.789 + 0.614i)2-s + (−0.213 − 0.976i)3-s + (0.245 + 0.969i)4-s + (0.922 + 0.386i)5-s + (0.431 − 0.901i)6-s + (−0.574 + 0.818i)7-s + (−0.401 + 0.915i)8-s + (−0.909 + 0.416i)9-s + (0.490 + 0.871i)10-s + (0.490 − 0.871i)11-s + (0.894 − 0.446i)12-s + (0.965 − 0.261i)13-s + (−0.956 + 0.293i)14-s + (0.180 − 0.983i)15-s + (−0.879 + 0.475i)16-s + (−0.724 − 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.932269986 + 1.133173499i\)
\(L(\frac12)\) \(\approx\) \(1.932269986 + 1.133173499i\)
\(L(1)\) \(\approx\) \(1.593770039 + 0.5018202762i\)
\(L(1)\) \(\approx\) \(1.593770039 + 0.5018202762i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.789 + 0.614i)T \)
3 \( 1 + (-0.213 - 0.976i)T \)
5 \( 1 + (0.922 + 0.386i)T \)
7 \( 1 + (-0.574 + 0.818i)T \)
11 \( 1 + (0.490 - 0.871i)T \)
13 \( 1 + (0.965 - 0.261i)T \)
17 \( 1 + (-0.724 - 0.689i)T \)
19 \( 1 + (0.746 + 0.665i)T \)
23 \( 1 + (0.863 + 0.504i)T \)
29 \( 1 + (0.546 + 0.837i)T \)
31 \( 1 + (-0.879 + 0.475i)T \)
37 \( 1 + (-0.627 - 0.778i)T \)
41 \( 1 + (-0.0825 + 0.996i)T \)
43 \( 1 + (0.980 - 0.197i)T \)
47 \( 1 + (-0.879 + 0.475i)T \)
53 \( 1 + (0.828 + 0.560i)T \)
59 \( 1 + (0.945 - 0.324i)T \)
61 \( 1 + (-0.999 - 0.0330i)T \)
67 \( 1 + (0.828 + 0.560i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.627 - 0.778i)T \)
79 \( 1 + (0.371 + 0.928i)T \)
83 \( 1 + (0.431 - 0.901i)T \)
89 \( 1 + (0.431 - 0.901i)T \)
97 \( 1 + (0.997 + 0.0660i)T \)
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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

−22.71514737444150392599934283813, −22.36239794368464098503763847914, −21.409116714843842944354699717889, −20.67334290683537700628362338934, −20.21787415699229640942340150678, −19.338587068701051074148582806578, −17.94081008868976625535234655804, −17.16181230161852312062800701075, −16.276384315544463103173929084350, −15.43927912234886055124041342812, −14.547175121267349787318047326166, −13.62193902298019352317000201714, −13.08789269246141441021647295856, −11.98504502695092996757996323189, −10.945555214774382749843743328703, −10.32273554762898892683373264518, −9.51297479602395357596811715776, −8.885264447848105660958952121306, −6.821692310543521345551577490826, −6.17951974942905844991123276894, −5.105547246920740073163880349550, −4.30610190937908441849118116294, −3.54970004967044765237357379738, −2.30368386181114094154476968846, −1.0034790135240817136498404985, 1.49480554545395107300799422554, 2.77143873182436822078878780489, 3.37093329004993313200484791901, 5.280099641053583258795924248338, 5.82458574607695086197739668781, 6.52117899799793094060682540377, 7.261302078737497426847755764669, 8.59243665574779828010377415888, 9.15713764011099123435259361375, 10.88818815061970647168435229095, 11.6096561267782195897129886564, 12.62465943440409549313262545090, 13.29876512861246628338069418445, 13.93918768411853055248626963108, 14.664254583369694303728350849764, 15.948235506815600211224133372759, 16.50348677106989710315796346050, 17.68160258323422070143973600558, 18.156996660584610064854119672156, 18.997505944225086492950015219426, 20.13829367485999851530439594145, 21.23488849623921753236849042656, 21.987684171647889998033461631755, 22.642448917486970694247167644121, 23.27205523410914930688384368622

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