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Properties

Label 1-571-571.222-r0-0-0
Degree $1$
Conductor $571$
Sign $0.956 - 0.292i$
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  + (−0.998 + 0.0550i)2-s + (−0.968 − 0.250i)3-s + (0.993 − 0.110i)4-s + (0.984 + 0.175i)5-s + (0.980 + 0.197i)6-s + (0.431 − 0.901i)7-s + (−0.986 + 0.164i)8-s + (0.874 + 0.485i)9-s + (−0.992 − 0.120i)10-s + (0.391 − 0.920i)11-s + (−0.989 − 0.142i)12-s + (0.287 + 0.957i)13-s + (−0.381 + 0.924i)14-s + (−0.909 − 0.416i)15-s + (0.975 − 0.218i)16-s + (0.509 + 0.860i)17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0550i)2-s + (−0.968 − 0.250i)3-s + (0.993 − 0.110i)4-s + (0.984 + 0.175i)5-s + (0.980 + 0.197i)6-s + (0.431 − 0.901i)7-s + (−0.986 + 0.164i)8-s + (0.874 + 0.485i)9-s + (−0.992 − 0.120i)10-s + (0.391 − 0.920i)11-s + (−0.989 − 0.142i)12-s + (0.287 + 0.957i)13-s + (−0.381 + 0.924i)14-s + (−0.909 − 0.416i)15-s + (0.975 − 0.218i)16-s + (0.509 + 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.956 - 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9185298935 - 0.1374955000i\)
\(L(\frac12)\) \(\approx\) \(0.9185298935 - 0.1374955000i\)
\(L(1)\) \(\approx\) \(0.7437060615 - 0.07128794661i\)
\(L(1)\) \(\approx\) \(0.7437060615 - 0.07128794661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.998 + 0.0550i)T \)
3 \( 1 + (-0.968 - 0.250i)T \)
5 \( 1 + (0.984 + 0.175i)T \)
7 \( 1 + (0.431 - 0.901i)T \)
11 \( 1 + (0.391 - 0.920i)T \)
13 \( 1 + (0.287 + 0.957i)T \)
17 \( 1 + (0.509 + 0.860i)T \)
19 \( 1 + (0.930 - 0.366i)T \)
23 \( 1 + (0.894 - 0.446i)T \)
29 \( 1 + (0.904 + 0.426i)T \)
31 \( 1 + (-0.677 - 0.735i)T \)
37 \( 1 + (-0.795 + 0.605i)T \)
41 \( 1 + (0.137 + 0.990i)T \)
43 \( 1 + (-0.421 + 0.906i)T \)
47 \( 1 + (-0.298 + 0.954i)T \)
53 \( 1 + (-0.256 - 0.966i)T \)
59 \( 1 + (-0.879 + 0.475i)T \)
61 \( 1 + (0.159 + 0.987i)T \)
67 \( 1 + (-0.709 + 0.705i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (-0.126 - 0.991i)T \)
79 \( 1 + (0.00551 - 0.999i)T \)
83 \( 1 + (-0.319 - 0.947i)T \)
89 \( 1 + (-0.660 + 0.750i)T \)
97 \( 1 + (0.202 - 0.979i)T \)
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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

−23.18964614306690093684870795733, −22.36682685797948624758146096058, −21.44093741523081475855209835276, −20.8657497171013326413430102760, −20.083184850112867464397967700036, −18.63050431664565269068192818334, −18.13121152918079581066506067839, −17.52379008789560585021520314629, −16.907310586468141150628013794231, −15.814402816570851241202626557254, −15.28844952981821284890573095827, −14.0891759045056311028426402994, −12.53433433100547591527391523578, −12.15753136761609920433972221417, −11.12560530948105058074963249057, −10.237659938256965365033477603963, −9.56001675583994361104611476909, −8.83695260567570407249490642221, −7.51068724080843823712112573366, −6.63530423095799881985837087280, −5.51817543214515128172112022053, −5.157985352685836767436507688791, −3.23982329145421334937660526088, −1.956781868415352105881720129794, −1.0439597827339317608852155474, 1.09030603742456790689159333396, 1.58616499310796824665190981040, 3.18092293731021483939885441421, 4.74391107792509754213222856684, 5.91309233397505044654091078214, 6.55038267995429245778201262452, 7.323525395182427422372728106664, 8.47229435318063937227148364473, 9.52824947836494719541148304621, 10.36833129046989289154983620716, 11.07097309351799138534458877915, 11.66803977951399646625994907456, 12.95356260994203840673699131699, 13.88787048005118949588072510445, 14.76341491054544033223709519937, 16.30677051442925292961093844706, 16.63562332549396118029417925433, 17.36844854935117328276360133406, 18.04916762818253344772097241031, 18.82035879164378120868160410859, 19.59600283503585562042081601626, 20.88693129076249099459258968001, 21.37869063513248230923072577936, 22.265263789515021414493892524136, 23.48349867898104749474072470131

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