Properties

Label 1-571-571.390-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.934 - 0.355i$
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 + (−0.401 + 0.915i)3-s + (0.245 + 0.969i)4-s + (−0.0825 + 0.996i)5-s + (−0.879 + 0.475i)6-s + (0.945 − 0.324i)7-s + (−0.401 + 0.915i)8-s + (−0.677 − 0.735i)9-s + (−0.677 + 0.735i)10-s + (−0.677 − 0.735i)11-s + (−0.986 − 0.164i)12-s + (0.546 + 0.837i)13-s + (0.945 + 0.324i)14-s + (−0.879 − 0.475i)15-s + (−0.879 + 0.475i)16-s + (−0.879 + 0.475i)17-s + ⋯
L(s)  = 1  + (0.789 + 0.614i)2-s + (−0.401 + 0.915i)3-s + (0.245 + 0.969i)4-s + (−0.0825 + 0.996i)5-s + (−0.879 + 0.475i)6-s + (0.945 − 0.324i)7-s + (−0.401 + 0.915i)8-s + (−0.677 − 0.735i)9-s + (−0.677 + 0.735i)10-s + (−0.677 − 0.735i)11-s + (−0.986 − 0.164i)12-s + (0.546 + 0.837i)13-s + (0.945 + 0.324i)14-s + (−0.879 − 0.475i)15-s + (−0.879 + 0.475i)16-s + (−0.879 + 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.934 - 0.355i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (390, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.934 - 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2823461992 + 1.534722856i\)
\(L(\frac12)\) \(\approx\) \(-0.2823461992 + 1.534722856i\)
\(L(1)\) \(\approx\) \(0.7505364266 + 1.087088941i\)
\(L(1)\) \(\approx\) \(0.7505364266 + 1.087088941i\)

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 \( 1 + (-0.401 + 0.915i)T \)
5 \( 1 + (-0.0825 + 0.996i)T \)
7 \( 1 + (0.945 - 0.324i)T \)
11 \( 1 + (-0.677 - 0.735i)T \)
13 \( 1 + (0.546 + 0.837i)T \)
17 \( 1 + (-0.879 + 0.475i)T \)
19 \( 1 + (-0.401 + 0.915i)T \)
23 \( 1 + (-0.401 - 0.915i)T \)
29 \( 1 + (0.546 + 0.837i)T \)
31 \( 1 + (-0.879 + 0.475i)T \)
37 \( 1 + (0.546 - 0.837i)T \)
41 \( 1 + (-0.0825 + 0.996i)T \)
43 \( 1 + (-0.677 + 0.735i)T \)
47 \( 1 + (-0.879 + 0.475i)T \)
53 \( 1 + (0.789 - 0.614i)T \)
59 \( 1 + (0.945 - 0.324i)T \)
61 \( 1 + (0.789 + 0.614i)T \)
67 \( 1 + (0.789 - 0.614i)T \)
71 \( 1 + T \)
73 \( 1 + (0.546 - 0.837i)T \)
79 \( 1 + (0.245 - 0.969i)T \)
83 \( 1 + (-0.879 + 0.475i)T \)
89 \( 1 + (-0.879 + 0.475i)T \)
97 \( 1 + (0.245 + 0.969i)T \)
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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

−22.93487075173577021840739128857, −21.96237568547173870690908083227, −21.075754337985297659688027585, −20.21876535410355941320116559125, −19.83518319114262215908218316193, −18.55565361337952145962655489815, −17.9043762424942163133874085112, −17.178827704497621984183124050016, −15.70445749376495970512947396677, −15.26280779420317855753840671477, −13.882034360322551843551938899067, −13.19534110728216444459274578254, −12.69686733837615123774355262245, −11.660429466936905241006604713893, −11.26647282866004997894562357871, −10.078674661508436115610948635800, −8.78601850704063580614279387889, −7.91999987189978013243439279350, −6.8448259393537445607399210555, −5.52900068405865695077725769342, −5.164691339805255795549657978960, −4.16818173059275868456871318108, −2.48692321904946548457677755130, −1.80489615547127182650798625130, −0.622222690891826509793379592234, 2.17949484534413876677665328626, 3.44100738970920899252083810933, 4.13547715835128397388605045968, 5.049021856450312987163390771832, 6.10311752500189397538548291762, 6.72822913392028442304637600706, 8.047968430041439974093715405876, 8.69424382730906988003979711622, 10.27487655347212688020858481579, 11.1106522952091391345386318428, 11.41810540729555104294433630531, 12.744330877566005312465198175189, 13.9781239208059889294647603590, 14.4988871835198708502850710860, 15.12453262982546785982019583743, 16.221829643873876289533347820887, 16.56628342198024556496476847804, 17.87744636973847980787388774760, 18.24262527786028750366828150205, 19.799388799335534358318099690854, 20.897567084769829963870921252478, 21.449548269012106705709158248916, 21.96959116196139200889136344505, 22.9878957390360801180772131534, 23.54504628245477214368532897144

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