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Properties

Label 1-571-571.80-r0-0-0
Degree $1$
Conductor $571$
Sign $0.874 - 0.485i$
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 + (−0.917 − 0.396i)3-s + (0.0275 − 0.999i)4-s + (−0.782 + 0.622i)5-s + (−0.934 + 0.355i)6-s + (−0.999 + 0.0330i)7-s + (−0.677 − 0.735i)8-s + (0.685 + 0.728i)9-s + (−0.126 + 0.991i)10-s + (−0.795 + 0.605i)11-s + (−0.421 + 0.906i)12-s + (0.583 − 0.812i)13-s + (−0.693 + 0.720i)14-s + (0.965 − 0.261i)15-s + (−0.998 − 0.0550i)16-s + (−0.360 + 0.932i)17-s + ⋯
L(s)  = 1  + (0.716 − 0.697i)2-s + (−0.917 − 0.396i)3-s + (0.0275 − 0.999i)4-s + (−0.782 + 0.622i)5-s + (−0.934 + 0.355i)6-s + (−0.999 + 0.0330i)7-s + (−0.677 − 0.735i)8-s + (0.685 + 0.728i)9-s + (−0.126 + 0.991i)10-s + (−0.795 + 0.605i)11-s + (−0.421 + 0.906i)12-s + (0.583 − 0.812i)13-s + (−0.693 + 0.720i)14-s + (0.965 − 0.261i)15-s + (−0.998 − 0.0550i)16-s + (−0.360 + 0.932i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8853162881 - 0.2291559145i\)
\(L(\frac12)\) \(\approx\) \(0.8853162881 - 0.2291559145i\)
\(L(1)\) \(\approx\) \(0.8104864747 - 0.3153955777i\)
\(L(1)\) \(\approx\) \(0.8104864747 - 0.3153955777i\)

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 \( 1 + (-0.917 - 0.396i)T \)
5 \( 1 + (-0.782 + 0.622i)T \)
7 \( 1 + (-0.999 + 0.0330i)T \)
11 \( 1 + (-0.795 + 0.605i)T \)
13 \( 1 + (0.583 - 0.812i)T \)
17 \( 1 + (-0.360 + 0.932i)T \)
19 \( 1 + (0.509 + 0.860i)T \)
23 \( 1 + (0.980 - 0.197i)T \)
29 \( 1 + (0.993 - 0.110i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (-0.949 - 0.314i)T \)
41 \( 1 + (0.350 + 0.936i)T \)
43 \( 1 + (-0.982 + 0.186i)T \)
47 \( 1 + (0.451 + 0.892i)T \)
53 \( 1 + (0.884 - 0.466i)T \)
59 \( 1 + (0.789 - 0.614i)T \)
61 \( 1 + (0.618 + 0.785i)T \)
67 \( 1 + (-0.0385 + 0.999i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.202 + 0.979i)T \)
79 \( 1 + (-0.997 - 0.0770i)T \)
83 \( 1 + (0.159 - 0.987i)T \)
89 \( 1 + (0.775 + 0.631i)T \)
97 \( 1 + (0.959 - 0.282i)T \)
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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

−23.183399070982755484099046812517, −22.84981754015489109409510333639, −21.76900522939236290515028140355, −21.151949503658936492366306647199, −20.24758207214125724692077302351, −19.060928429230304150647379214700, −18.11615836628443637359422411127, −17.03396099941115381000889536911, −16.33334883719491082494607922810, −15.747219941596515651222282356204, −15.46276073928366177866896008279, −13.766127301607419928112916492179, −13.19925734780749680866704139439, −12.193814200605113530985525053014, −11.62144227242410041773501994368, −10.6744112435231760912438216745, −9.24172857511091682155897667194, −8.574941275171279732326687595068, −7.101141320036225734975516281840, −6.690616674929658626771364746160, −5.39653862602557197161188606944, −4.869853006359515160890661018285, −3.811684430473046863898189568098, −2.98460369356848329999597772911, −0.584481414032833872421730591914, 0.8884586087689805609804663493, 2.42030891597228177895084559181, 3.3829744038611009952431599223, 4.358540849288571222204190466967, 5.492093027407308103502916439452, 6.319059803794847221399025621345, 7.08353221408892288851502786279, 8.24507337939873407173467589596, 10.020549048966822083941670852305, 10.41385915501158264383902757515, 11.27253431669560744186652756441, 12.17567259221944564692445117140, 12.8413988277736654192200843552, 13.41312002254551891116761893955, 14.79501517758620591302608823391, 15.61342231729706182303217888445, 16.128990394378300056823872889059, 17.55845749500087008574596347052, 18.461758636225494351755369583304, 19.02342266722395294113461770853, 19.76551203059768881846330825974, 20.747674990306648039218927937721, 21.768887940495443713029746654292, 22.71093893804633366742485645983, 22.937172600275513723545758836042

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