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Properties

Label 1-571-571.10-r1-0-0
Degree $1$
Conductor $571$
Sign $0.605 + 0.795i$
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  + (−0.851 − 0.523i)2-s + (−0.329 − 0.944i)3-s + (0.451 + 0.892i)4-s + (0.874 − 0.485i)5-s + (−0.213 + 0.976i)6-s + (0.768 − 0.639i)7-s + (0.0825 − 0.996i)8-s + (−0.782 + 0.622i)9-s + (−0.999 − 0.0440i)10-s + (−0.537 − 0.843i)11-s + (0.693 − 0.720i)12-s + (0.509 + 0.860i)13-s + (−0.989 + 0.142i)14-s + (−0.746 − 0.665i)15-s + (−0.592 + 0.805i)16-s + (−0.583 − 0.812i)17-s + ⋯
L(s)  = 1  + (−0.851 − 0.523i)2-s + (−0.329 − 0.944i)3-s + (0.451 + 0.892i)4-s + (0.874 − 0.485i)5-s + (−0.213 + 0.976i)6-s + (0.768 − 0.639i)7-s + (0.0825 − 0.996i)8-s + (−0.782 + 0.622i)9-s + (−0.999 − 0.0440i)10-s + (−0.537 − 0.843i)11-s + (0.693 − 0.720i)12-s + (0.509 + 0.860i)13-s + (−0.989 + 0.142i)14-s + (−0.746 − 0.665i)15-s + (−0.592 + 0.805i)16-s + (−0.583 − 0.812i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2823604833 + 0.1400078312i\)
\(L(\frac12)\) \(\approx\) \(0.2823604833 + 0.1400078312i\)
\(L(1)\) \(\approx\) \(0.5676235715 - 0.3419634545i\)
\(L(1)\) \(\approx\) \(0.5676235715 - 0.3419634545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.851 - 0.523i)T \)
3 \( 1 + (-0.329 - 0.944i)T \)
5 \( 1 + (0.874 - 0.485i)T \)
7 \( 1 + (0.768 - 0.639i)T \)
11 \( 1 + (-0.537 - 0.843i)T \)
13 \( 1 + (0.509 + 0.860i)T \)
17 \( 1 + (-0.583 - 0.812i)T \)
19 \( 1 + (-0.287 + 0.957i)T \)
23 \( 1 + (-0.518 + 0.854i)T \)
29 \( 1 + (-0.298 + 0.954i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (0.0935 + 0.995i)T \)
41 \( 1 + (0.191 + 0.981i)T \)
43 \( 1 + (0.266 + 0.963i)T \)
47 \( 1 + (-0.993 - 0.110i)T \)
53 \( 1 + (0.234 + 0.972i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (-0.609 + 0.792i)T \)
67 \( 1 + (-0.959 - 0.282i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.815 + 0.578i)T \)
79 \( 1 + (-0.840 - 0.542i)T \)
83 \( 1 + (-0.739 - 0.673i)T \)
89 \( 1 + (-0.952 + 0.303i)T \)
97 \( 1 + (-0.709 + 0.705i)T \)
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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

−22.85766977445981766006109477585, −22.1050008919331381955045640367, −21.1004944393136086912722613559, −20.60864838415556644206238211337, −19.55306381291196546098361246679, −18.25280949890414699842409616626, −17.740145253426327125562165107258, −17.38070896218777956100384917192, −16.16281558518251911888293220180, −15.19787884821393167594386163264, −15.01504275680067904962574381209, −13.95978077171950715917914878171, −12.610013606231612757840791400803, −11.25275712443557131078180682427, −10.63228382844754210485439510345, −10.05633632939821786409792962933, −9.00749548927647571183496594393, −8.40110253985615289812425856097, −7.11324195471458837052410696570, −5.996854245993273210944457925172, −5.46234508750625529391803731737, −4.46518795146174989313417139862, −2.6919550264575286952954249881, −1.84184359309103133100815432859, −0.104602309727305912312181186849, 1.24350053664732735717575125986, 1.68577183515107951845450419269, 2.881265001079282421755390951235, 4.396648438197627648730138876703, 5.68825936693416287715060383095, 6.595927851360750667148967191895, 7.65686943919404540257220216634, 8.35296489527298098827314631262, 9.236596870943461336740046971152, 10.354532926296598145087233444954, 11.24906747654672836124389672476, 11.7615356953364273776022344320, 13.0636352379915553313677030588, 13.474241588698805182403351034530, 14.36069038837761511432363979534, 16.2473860351475172958730533685, 16.636995147161265081254287429778, 17.51111745895538866881346719100, 18.239817955940745180100879010726, 18.68070544492012168785149627565, 19.8405902026166712041527896163, 20.56938000344877749754972918745, 21.29061566536713808429685056315, 22.07618978782410787249486944054, 23.38471159000027322726269827936

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