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Properties

Label 1-571-571.125-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: $\chi_{571} (125, \cdot )$
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

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

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