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Properties

Label 1-571-571.97-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.457 - 0.889i$
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  + (0.350 + 0.936i)2-s + (−0.997 − 0.0770i)3-s + (−0.754 + 0.656i)4-s + (0.411 + 0.911i)5-s + (−0.277 − 0.960i)6-s + (−0.518 + 0.854i)7-s + (−0.879 − 0.475i)8-s + (0.988 + 0.153i)9-s + (−0.709 + 0.705i)10-s + (−0.256 + 0.966i)11-s + (0.802 − 0.596i)12-s + (−0.441 + 0.897i)13-s + (−0.982 − 0.186i)14-s + (−0.340 − 0.940i)15-s + (0.137 − 0.990i)16-s + (−0.899 − 0.436i)17-s + ⋯
L(s)  = 1  + (0.350 + 0.936i)2-s + (−0.997 − 0.0770i)3-s + (−0.754 + 0.656i)4-s + (0.411 + 0.911i)5-s + (−0.277 − 0.960i)6-s + (−0.518 + 0.854i)7-s + (−0.879 − 0.475i)8-s + (0.988 + 0.153i)9-s + (−0.709 + 0.705i)10-s + (−0.256 + 0.966i)11-s + (0.802 − 0.596i)12-s + (−0.441 + 0.897i)13-s + (−0.982 − 0.186i)14-s + (−0.340 − 0.940i)15-s + (0.137 − 0.990i)16-s + (−0.899 − 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.457 - 0.889i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.457 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3713025218 + 0.6088543531i\)
\(L(\frac12)\) \(\approx\) \(-0.3713025218 + 0.6088543531i\)
\(L(1)\) \(\approx\) \(0.4157571406 + 0.6444321463i\)
\(L(1)\) \(\approx\) \(0.4157571406 + 0.6444321463i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.350 + 0.936i)T \)
3 \( 1 + (-0.997 - 0.0770i)T \)
5 \( 1 + (0.411 + 0.911i)T \)
7 \( 1 + (-0.518 + 0.854i)T \)
11 \( 1 + (-0.256 + 0.966i)T \)
13 \( 1 + (-0.441 + 0.897i)T \)
17 \( 1 + (-0.899 - 0.436i)T \)
19 \( 1 + (0.761 + 0.648i)T \)
23 \( 1 + (0.991 + 0.131i)T \)
29 \( 1 + (-0.962 + 0.272i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.884 + 0.466i)T \)
41 \( 1 + (0.993 - 0.110i)T \)
43 \( 1 + (-0.889 - 0.456i)T \)
47 \( 1 + (-0.926 + 0.376i)T \)
53 \( 1 + (-0.782 - 0.622i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (-0.968 + 0.250i)T \)
67 \( 1 + (0.930 - 0.366i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (-0.0385 - 0.999i)T \)
79 \( 1 + (0.731 - 0.681i)T \)
83 \( 1 + (0.970 + 0.240i)T \)
89 \( 1 + (-0.693 + 0.720i)T \)
97 \( 1 + (-0.857 - 0.514i)T \)
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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

−22.63022007817696972441721878561, −21.81797366415719735925452889932, −21.163077426462390137758107479074, −20.22169744094861772406503561051, −19.61316264329814672574579188997, −18.532369953522558034757760991779, −17.57550688904747602598219196804, −17.02206980427666739376799502403, −16.10381942683724926025475280127, −15.12858646657045366750957892011, −13.63604779948017577533730692817, −13.12787620484522704450852536377, −12.6188356610287065731863218734, −11.37778262455957513772911274074, −10.85966073643064493150312826551, −9.88457933300825253407636139495, −9.24927472557874631926818132238, −7.92002408330845508054685089336, −6.46432128936624576614361248441, −5.60717591838122802781504303996, −4.83767380396369050381210426667, −3.968793017979599791639704227298, −2.73646025859943350245121178265, −1.17218200267487203738264364525, −0.42044564965067354963584086504, 2.01985604455234883732094874465, 3.257660953589013696977075806578, 4.65103905756763374123198062614, 5.340612062348142388612948546614, 6.42456508744359616415499250651, 6.80594661032157783097530890124, 7.7117756633025658174338189695, 9.37612652864325475966802797064, 9.72986419170031503032388149963, 11.12962234618532525633615551231, 12.01328486897356551741749191087, 12.796185352174155094403317778366, 13.66001832324278426320056150408, 14.7654444545808955599866021801, 15.39928964399126617937856515900, 16.210953202529076851131621066975, 17.07430451507933913145034899085, 17.961864889641598846349304021116, 18.383122540097019333335322196822, 19.2169108556052126280702995060, 20.95782234283000330198903019196, 21.75275570578746467209968841059, 22.44169216276385926449647981953, 22.794845063896676491929472435686, 23.6830234886364309611869098208

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