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Properties

Label 1-571-571.50-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: $\chi_{571} (50, \cdot )$
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

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

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