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Properties

Label 1-571-571.162-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.218 - 0.975i$
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 + (0.973 − 0.229i)3-s + (0.546 − 0.837i)4-s + (−0.956 + 0.293i)5-s + (0.746 − 0.665i)6-s + (−0.997 + 0.0660i)7-s + (0.0825 − 0.996i)8-s + (0.894 − 0.446i)9-s + (−0.701 + 0.712i)10-s + (0.701 + 0.712i)11-s + (0.340 − 0.940i)12-s + (0.980 + 0.197i)13-s + (−0.846 + 0.533i)14-s + (−0.863 + 0.504i)15-s + (−0.401 − 0.915i)16-s + (0.213 − 0.976i)17-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)2-s + (0.973 − 0.229i)3-s + (0.546 − 0.837i)4-s + (−0.956 + 0.293i)5-s + (0.746 − 0.665i)6-s + (−0.997 + 0.0660i)7-s + (0.0825 − 0.996i)8-s + (0.894 − 0.446i)9-s + (−0.701 + 0.712i)10-s + (0.701 + 0.712i)11-s + (0.340 − 0.940i)12-s + (0.980 + 0.197i)13-s + (−0.846 + 0.533i)14-s + (−0.863 + 0.504i)15-s + (−0.401 − 0.915i)16-s + (0.213 − 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.218 - 0.975i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.218 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.654712739 - 3.314755648i\)
\(L(\frac12)\) \(\approx\) \(2.654712739 - 3.314755648i\)
\(L(1)\) \(\approx\) \(1.896996441 - 0.9708391091i\)
\(L(1)\) \(\approx\) \(1.896996441 - 0.9708391091i\)

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 \( 1 + (0.973 - 0.229i)T \)
5 \( 1 + (-0.956 + 0.293i)T \)
7 \( 1 + (-0.997 + 0.0660i)T \)
11 \( 1 + (0.701 + 0.712i)T \)
13 \( 1 + (0.980 + 0.197i)T \)
17 \( 1 + (0.213 - 0.976i)T \)
19 \( 1 + (0.518 + 0.854i)T \)
23 \( 1 + (0.922 - 0.386i)T \)
29 \( 1 + (-0.677 - 0.735i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (0.115 - 0.993i)T \)
41 \( 1 + (-0.945 - 0.324i)T \)
43 \( 1 + (-0.148 + 0.988i)T \)
47 \( 1 + (0.401 + 0.915i)T \)
53 \( 1 + (-0.431 - 0.901i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (-0.724 - 0.689i)T \)
67 \( 1 + (-0.431 - 0.901i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.115 + 0.993i)T \)
79 \( 1 + (0.627 - 0.778i)T \)
83 \( 1 + (0.746 - 0.665i)T \)
89 \( 1 + (-0.746 + 0.665i)T \)
97 \( 1 + (0.0495 + 0.998i)T \)
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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

−23.5228553639526821382648720473, −22.33766925233265708595056308783, −21.81226365491080506475611428442, −20.77322736514350717041578302242, −19.97294806307747952883659107167, −19.46985663374407534047832942164, −18.52653820841399511825250049126, −16.84380754966817907000641945438, −16.34738284881044149021175460593, −15.425267732839334699349325675863, −15.08980986365895273412059379461, −13.85830922097424203893297182774, −13.25768844327553403764189233375, −12.52132575594484969260180873908, −11.45759985890142924266161324270, −10.50897311369172560691102554060, −8.86862390565780032313803821961, −8.65370908283335501698174652054, −7.39703829504843706169923606905, −6.72796014862536864040624350680, −5.506974162140728084511824278786, −4.286448518083066746232754252072, −3.37984932095911415557612072357, −3.1983830771784446939564281662, −1.29854888765347884534783626005, 0.71674270363340034103560653859, 2.016102790462786272596878166788, 3.224922333469197390782713127942, 3.63903567779034506580018218889, 4.57078147756394683215428836733, 6.15290097847313174135391717302, 6.95849979504948210060346926483, 7.73240278834421910246952651380, 9.17864559754155963314599823370, 9.74699436688955754408404381323, 10.97195370686423543614857040909, 11.91942501752693403761737495463, 12.62952513428184415666014325906, 13.42119211281549316021721940284, 14.32030914702139338522351701412, 15.01392362132143751357729375172, 15.78563377236341521462129603104, 16.45586851394561457413969189374, 18.40742401083898100206079336442, 18.90749939875598476973386222453, 19.57582615526694447461806594854, 20.45225537954411725226246570697, 20.783600341837676104754816569975, 22.18229793525741971813469220223, 22.846925438315469698226101505249

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