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Properties

Label 1-571-571.217-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.441 + 0.897i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (0.879 − 0.475i)2-s + (−0.922 − 0.386i)3-s + (0.546 − 0.837i)4-s + (−0.574 − 0.818i)5-s + (−0.995 + 0.0990i)6-s + (0.846 + 0.533i)7-s + (0.0825 − 0.996i)8-s + (0.701 + 0.712i)9-s + (−0.894 − 0.446i)10-s + (0.894 − 0.446i)11-s + (−0.828 + 0.560i)12-s + (0.115 + 0.993i)13-s + (0.997 + 0.0660i)14-s + (0.213 + 0.976i)15-s + (−0.401 − 0.915i)16-s + (−0.863 − 0.504i)17-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)2-s + (−0.922 − 0.386i)3-s + (0.546 − 0.837i)4-s + (−0.574 − 0.818i)5-s + (−0.995 + 0.0990i)6-s + (0.846 + 0.533i)7-s + (0.0825 − 0.996i)8-s + (0.701 + 0.712i)9-s + (−0.894 − 0.446i)10-s + (0.894 − 0.446i)11-s + (−0.828 + 0.560i)12-s + (0.115 + 0.993i)13-s + (0.997 + 0.0660i)14-s + (0.213 + 0.976i)15-s + (−0.401 − 0.915i)16-s + (−0.863 − 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3202614752 - 0.5145919798i\)
\(L(\frac12)\) \(\approx\) \(-0.3202614752 - 0.5145919798i\)
\(L(1)\) \(\approx\) \(0.9041513259 - 0.6214429562i\)
\(L(1)\) \(\approx\) \(0.9041513259 - 0.6214429562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.879 - 0.475i)T \)
3 \( 1 + (-0.922 - 0.386i)T \)
5 \( 1 + (-0.574 - 0.818i)T \)
7 \( 1 + (0.846 + 0.533i)T \)
11 \( 1 + (0.894 - 0.446i)T \)
13 \( 1 + (0.115 + 0.993i)T \)
17 \( 1 + (-0.863 - 0.504i)T \)
19 \( 1 + (-0.652 + 0.757i)T \)
23 \( 1 + (-0.973 - 0.229i)T \)
29 \( 1 + (-0.677 - 0.735i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (0.980 - 0.197i)T \)
41 \( 1 + (-0.945 - 0.324i)T \)
43 \( 1 + (-0.461 - 0.887i)T \)
47 \( 1 + (0.401 + 0.915i)T \)
53 \( 1 + (-0.991 + 0.131i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (0.180 + 0.983i)T \)
67 \( 1 + (-0.991 + 0.131i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.980 + 0.197i)T \)
79 \( 1 + (-0.965 + 0.261i)T \)
83 \( 1 + (-0.995 + 0.0990i)T \)
89 \( 1 + (0.995 - 0.0990i)T \)
97 \( 1 + (-0.934 + 0.355i)T \)
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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

−23.58879887568025888845971026350, −22.796181457011241743215780053486, −22.01182980733011016346294290603, −21.664848408774257477026857891099, −20.29534861763875545725629099865, −19.84211824812803571300080001997, −18.094676860124822319153504485972, −17.656854953600703110961302271008, −16.84342916124888842022644963357, −15.87672988122626068777251632144, −14.96587511249500903184188504976, −14.739215779793475605913760370708, −13.44475491096753439520937102792, −12.46985915974507534051396432383, −11.55021439666025973813058036732, −11.033857043219185931928890230054, −10.25989471189516960354532882269, −8.614051264507117069321735027152, −7.52382994190699777841378916330, −6.7916596950397254732849971418, −6.02655522164174845459330480769, −4.809436937879957326870796318543, −4.188373668734233367559757727178, −3.320537075006381990040548250900, −1.72190725175290522201539054412, 0.11779930697467510176481504081, 1.41950211892695884167169332357, 2.10700494290164416728935100477, 4.126315427286674599375819942306, 4.388619295917459392083614594115, 5.5821310227404494170722681726, 6.23732141727256954822601602265, 7.3657445203720198389833022767, 8.542634287331129126231511745771, 9.60803729339804675420387870466, 11.01389829843099996810636312932, 11.6218750801571649924131067393, 11.97150536180108077595616428038, 12.92565691045579568759511609143, 13.8109990253589695189653386838, 14.75530280415617886118335360636, 15.77543875497450803387497211003, 16.51010489793567300803889396117, 17.30895042962976154076937761021, 18.636479619106802823261817475692, 19.04861541324763741873178777392, 20.18634466545362632212037602178, 20.90633060069341652102084732162, 21.87877899990460070483686757306, 22.32130334689461782130002520857

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