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Properties

Label 1-571-571.133-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.928 - 0.370i$
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  + (0.451 − 0.892i)2-s + (0.266 + 0.963i)3-s + (−0.592 − 0.805i)4-s + (−0.644 + 0.764i)5-s + (0.980 + 0.197i)6-s + (0.431 − 0.901i)7-s + (−0.986 + 0.164i)8-s + (−0.857 + 0.514i)9-s + (0.391 + 0.920i)10-s + (−0.992 + 0.120i)11-s + (0.618 − 0.785i)12-s + (0.685 − 0.728i)13-s + (−0.609 − 0.792i)14-s + (−0.909 − 0.416i)15-s + (−0.298 + 0.954i)16-s + (−0.999 + 0.0110i)17-s + ⋯
L(s)  = 1  + (0.451 − 0.892i)2-s + (0.266 + 0.963i)3-s + (−0.592 − 0.805i)4-s + (−0.644 + 0.764i)5-s + (0.980 + 0.197i)6-s + (0.431 − 0.901i)7-s + (−0.986 + 0.164i)8-s + (−0.857 + 0.514i)9-s + (0.391 + 0.920i)10-s + (−0.992 + 0.120i)11-s + (0.618 − 0.785i)12-s + (0.685 − 0.728i)13-s + (−0.609 − 0.792i)14-s + (−0.909 − 0.416i)15-s + (−0.298 + 0.954i)16-s + (−0.999 + 0.0110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.928 - 0.370i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.928 - 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1125454830 - 0.5854005630i\)
\(L(\frac12)\) \(\approx\) \(0.1125454830 - 0.5854005630i\)
\(L(1)\) \(\approx\) \(0.8501005170 - 0.3019246123i\)
\(L(1)\) \(\approx\) \(0.8501005170 - 0.3019246123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.451 - 0.892i)T \)
3 \( 1 + (0.266 + 0.963i)T \)
5 \( 1 + (-0.644 + 0.764i)T \)
7 \( 1 + (0.431 - 0.901i)T \)
11 \( 1 + (-0.992 + 0.120i)T \)
13 \( 1 + (0.685 - 0.728i)T \)
17 \( 1 + (-0.999 + 0.0110i)T \)
19 \( 1 + (-0.782 - 0.622i)T \)
23 \( 1 + (0.894 - 0.446i)T \)
29 \( 1 + (-0.821 + 0.569i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.126 - 0.991i)T \)
41 \( 1 + (-0.926 - 0.376i)T \)
43 \( 1 + (0.996 - 0.0880i)T \)
47 \( 1 + (0.975 - 0.218i)T \)
53 \( 1 + (-0.709 + 0.705i)T \)
59 \( 1 + (-0.879 + 0.475i)T \)
61 \( 1 + (0.775 - 0.631i)T \)
67 \( 1 + (-0.256 - 0.966i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.795 + 0.605i)T \)
79 \( 1 + (-0.868 + 0.495i)T \)
83 \( 1 + (-0.660 + 0.750i)T \)
89 \( 1 + (-0.319 - 0.947i)T \)
97 \( 1 + (-0.949 + 0.314i)T \)
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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

−23.677355200593407660300174175794, −23.3243279777908454142151326724, −22.11593468923174616692421414703, −21.02542967746529353832680608082, −20.55449472311524934223548209399, −19.07004655454120572186691809480, −18.65708257551307892022769907108, −17.698943141398249037893191333769, −16.884637070669116451347074899245, −15.84510665895325146700530239004, −15.2582330813148955406288671244, −14.36084611208794866209124312443, −13.20488473358872712913934535504, −12.92092705093410649814886737921, −11.913634145979969528240427920713, −11.21767891919173275676962346615, −9.09866750620478151521700893541, −8.61436785556339542858319178345, −7.95443743151588889643029479315, −7.03677361879855365381877206689, −5.98992997870978513057129283929, −5.19324184302960698919581374643, −4.13989881143889181833232201212, −2.919999926830997567535000674703, −1.66424750542717432989131052483, 0.24310645576356331314267046856, 2.22493563973693807739321704501, 3.1325674247821840720445837649, 3.997461319436489559546287321412, 4.66593407775363209514009013189, 5.73737976835100977563254982920, 7.15714309783693033928878223519, 8.27476627250205832875312922072, 9.18437325559965669314667965926, 10.55664265694925972411761550821, 10.75019124240803583953360011532, 11.24328930517240378673451181217, 12.77205408802545666572879929389, 13.5336197610461338283460146736, 14.44009956129641554837230081850, 15.23696452488219263213255060166, 15.69099744950249062060813533974, 17.09242342255425274338288596703, 18.06683549137656850519968144786, 18.91568422968210913391358337047, 19.948225273014310628803994616172, 20.39150140013168634720265323596, 21.11585925969330259499440782567, 22.041981801583406511835378753051, 22.75264309289718832358419448934

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