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Properties

Label 1-571-571.201-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.973 - 0.227i$
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  + (−0.986 − 0.164i)2-s + (0.180 − 0.983i)3-s + (0.945 + 0.324i)4-s + (−0.213 − 0.976i)5-s + (−0.340 + 0.940i)6-s + (0.922 − 0.386i)7-s + (−0.879 − 0.475i)8-s + (−0.934 − 0.355i)9-s + (0.0495 + 0.998i)10-s + (0.0495 − 0.998i)11-s + (0.490 − 0.871i)12-s + (0.371 + 0.928i)13-s + (−0.973 + 0.229i)14-s + (−0.999 + 0.0330i)15-s + (0.789 + 0.614i)16-s + (−0.277 − 0.960i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.164i)2-s + (0.180 − 0.983i)3-s + (0.945 + 0.324i)4-s + (−0.213 − 0.976i)5-s + (−0.340 + 0.940i)6-s + (0.922 − 0.386i)7-s + (−0.879 − 0.475i)8-s + (−0.934 − 0.355i)9-s + (0.0495 + 0.998i)10-s + (0.0495 − 0.998i)11-s + (0.490 − 0.871i)12-s + (0.371 + 0.928i)13-s + (−0.973 + 0.229i)14-s + (−0.999 + 0.0330i)15-s + (0.789 + 0.614i)16-s + (−0.277 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.973 - 0.227i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1021493201 - 0.8874939725i\)
\(L(\frac12)\) \(\approx\) \(0.1021493201 - 0.8874939725i\)
\(L(1)\) \(\approx\) \(0.5666130916 - 0.5250309508i\)
\(L(1)\) \(\approx\) \(0.5666130916 - 0.5250309508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.986 - 0.164i)T \)
3 \( 1 + (0.180 - 0.983i)T \)
5 \( 1 + (-0.213 - 0.976i)T \)
7 \( 1 + (0.922 - 0.386i)T \)
11 \( 1 + (0.0495 - 0.998i)T \)
13 \( 1 + (0.371 + 0.928i)T \)
17 \( 1 + (-0.277 - 0.960i)T \)
19 \( 1 + (0.991 - 0.131i)T \)
23 \( 1 + (-0.724 - 0.689i)T \)
29 \( 1 + (0.245 - 0.969i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (-0.768 - 0.639i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (-0.627 - 0.778i)T \)
47 \( 1 + (0.789 + 0.614i)T \)
53 \( 1 + (0.894 + 0.446i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (-0.148 - 0.988i)T \)
67 \( 1 + (0.894 + 0.446i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.768 - 0.639i)T \)
79 \( 1 + (0.601 + 0.799i)T \)
83 \( 1 + (-0.340 + 0.940i)T \)
89 \( 1 + (-0.340 + 0.940i)T \)
97 \( 1 + (-0.956 + 0.293i)T \)
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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

−23.67473585995938028388709484172, −22.6848850437168431214656232258, −21.88398099892677914841606873684, −20.9796865024958400632250184506, −20.20368762902330531261308800573, −19.593490715138798057424217895688, −18.378750086135704482527995661950, −17.81175398737436509486167904120, −17.14650167205176639043175380312, −15.84995510616176911054292853666, −15.24652160337169795296507526386, −14.86484612578099356486472380484, −13.86535655361964030386162886858, −12.03268524043421722954198646270, −11.40747275471831759104994901645, −10.35161962419167126154834463122, −10.14621857685983203476189419033, −8.87534853608625502884892129288, −8.07756558205139921971600553828, −7.33236536090292002611414514443, −6.04380279249198247047287996970, −5.16974523340462800602589152674, −3.78187926009678820815966150685, −2.75622415095153535542681825316, −1.7080406627830659359465292045, 0.660207351431721701924233293488, 1.41534756053999416174964679269, 2.4886330494811145397830501092, 3.849295808911074092422068521088, 5.25687460607254607530867697942, 6.42603410975533709374095861568, 7.35891477440577266299321675291, 8.26472636008723614264274284629, 8.64921615025337089744364392808, 9.64126478592893010083390001665, 11.107199390628689102883918161223, 11.67626385289512299674287587908, 12.28460771705939826355534173465, 13.6752343729669487214973342047, 14.032008165511672316587349160600, 15.608220089945431513273691238024, 16.412911104866542905282261742658, 17.13066166031770573648621247562, 17.94628352779158916279225710508, 18.66011290214522153786829719535, 19.444886931890247458583490057008, 20.328673126841765136072867335498, 20.73117900169961986647938139059, 21.7179652779094210455506299446, 23.27988941142575258897055355766

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