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Properties

Label 1-571-571.244-r0-0-0
Degree $1$
Conductor $571$
Sign $0.997 - 0.0739i$
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  + (0.716 + 0.697i)2-s + (0.509 − 0.860i)3-s + (0.0275 + 0.999i)4-s + (−0.834 + 0.551i)5-s + (0.965 − 0.261i)6-s + (0.828 − 0.560i)7-s + (−0.677 + 0.735i)8-s + (−0.480 − 0.876i)9-s + (−0.982 − 0.186i)10-s + (0.329 − 0.944i)11-s + (0.874 + 0.485i)12-s + (0.952 − 0.303i)13-s + (0.984 + 0.175i)14-s + (0.0495 + 0.998i)15-s + (−0.998 + 0.0550i)16-s + (0.775 − 0.631i)17-s + ⋯
L(s)  = 1  + (0.716 + 0.697i)2-s + (0.509 − 0.860i)3-s + (0.0275 + 0.999i)4-s + (−0.834 + 0.551i)5-s + (0.965 − 0.261i)6-s + (0.828 − 0.560i)7-s + (−0.677 + 0.735i)8-s + (−0.480 − 0.876i)9-s + (−0.982 − 0.186i)10-s + (0.329 − 0.944i)11-s + (0.874 + 0.485i)12-s + (0.952 − 0.303i)13-s + (0.984 + 0.175i)14-s + (0.0495 + 0.998i)15-s + (−0.998 + 0.0550i)16-s + (0.775 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.997 - 0.0739i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.997 - 0.0739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.278425566 - 0.08431982937i\)
\(L(\frac12)\) \(\approx\) \(2.278425566 - 0.08431982937i\)
\(L(1)\) \(\approx\) \(1.695383404 + 0.1339902257i\)
\(L(1)\) \(\approx\) \(1.695383404 + 0.1339902257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.716 + 0.697i)T \)
3 \( 1 + (0.509 - 0.860i)T \)
5 \( 1 + (-0.834 + 0.551i)T \)
7 \( 1 + (0.828 - 0.560i)T \)
11 \( 1 + (0.329 - 0.944i)T \)
13 \( 1 + (0.952 - 0.303i)T \)
17 \( 1 + (0.775 - 0.631i)T \)
19 \( 1 + (-0.660 - 0.750i)T \)
23 \( 1 + (-0.909 + 0.416i)T \)
29 \( 1 + (0.993 + 0.110i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (0.00551 + 0.999i)T \)
41 \( 1 + (0.350 - 0.936i)T \)
43 \( 1 + (0.685 + 0.728i)T \)
47 \( 1 + (0.451 - 0.892i)T \)
53 \( 1 + (-0.170 + 0.985i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.0385 + 0.999i)T \)
67 \( 1 + (0.938 - 0.345i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.868 - 0.495i)T \)
79 \( 1 + (0.761 - 0.648i)T \)
83 \( 1 + (-0.709 - 0.705i)T \)
89 \( 1 + (-0.256 + 0.966i)T \)
97 \( 1 + (0.565 - 0.824i)T \)
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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

−23.15564849170555255526668362626, −22.35208364225596626618997979415, −21.27358071201079380973411155985, −20.8928155726760358165645771874, −20.228729106357646493120157554869, −19.35638503864880259928121475366, −18.65395494598102179403734459686, −17.35329043190101523786419058240, −16.110871946969067561757713868, −15.53798629730760655355686262810, −14.62446452350289951407041004129, −14.25494359156043534856880804629, −12.87658218493529481681356390577, −12.117214443560334916519140722, −11.36393064974419987242153063589, −10.48706079067105168821199784279, −9.5534579230645401673977326661, −8.56749548873304088476606739199, −7.89677334404624915811411187130, −6.15795970347568467753352326167, −5.15523021205963148888538731857, −4.155850485012062690562502954460, −3.926849801801028217626900366499, −2.45045142558090255581221714028, −1.45950236373304388240698369651, 1.00331457657206357241207838722, 2.71051266137730542529467996415, 3.50150704775617599257008032249, 4.372382952723557817398053633203, 5.77253632567893342416187219630, 6.67647336415192654656269712219, 7.44312127831120254197181283132, 8.22684415591806346827564624013, 8.71825725684995354374022699218, 10.65986715127941442225377595354, 11.61489498537656097459751011189, 12.12096859795955347116415212711, 13.443358463535170978464277773522, 13.93871742948610397161397574571, 14.58149321639137966577442818694, 15.51038046827662135806025735385, 16.30732255067449297049319260787, 17.47655595349137058174908863630, 18.11518956694500202200590844871, 19.049526676586196307780165553607, 19.95140908349533853499745983152, 20.78971661637874525752887946361, 21.633636345769995552403697935810, 22.782015285794288681012936085924, 23.55442603687976464804669333549

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