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Properties

Label 1-571-571.359-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.410 + 0.912i$
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  + (−0.821 − 0.569i)2-s + (0.618 − 0.785i)3-s + (0.350 + 0.936i)4-s + (−0.256 + 0.966i)5-s + (−0.956 + 0.293i)6-s + (0.115 − 0.993i)7-s + (0.245 − 0.969i)8-s + (−0.234 − 0.972i)9-s + (0.761 − 0.648i)10-s + (−0.942 + 0.335i)11-s + (0.952 + 0.303i)12-s + (−0.644 + 0.764i)13-s + (−0.660 + 0.750i)14-s + (0.601 + 0.799i)15-s + (−0.754 + 0.656i)16-s + (−0.857 − 0.514i)17-s + ⋯
L(s)  = 1  + (−0.821 − 0.569i)2-s + (0.618 − 0.785i)3-s + (0.350 + 0.936i)4-s + (−0.256 + 0.966i)5-s + (−0.956 + 0.293i)6-s + (0.115 − 0.993i)7-s + (0.245 − 0.969i)8-s + (−0.234 − 0.972i)9-s + (0.761 − 0.648i)10-s + (−0.942 + 0.335i)11-s + (0.952 + 0.303i)12-s + (−0.644 + 0.764i)13-s + (−0.660 + 0.750i)14-s + (0.601 + 0.799i)15-s + (−0.754 + 0.656i)16-s + (−0.857 − 0.514i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.410 + 0.912i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.410 + 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03938041770 + 0.06089037351i\)
\(L(\frac12)\) \(\approx\) \(0.03938041770 + 0.06089037351i\)
\(L(1)\) \(\approx\) \(0.5564543330 - 0.1983301850i\)
\(L(1)\) \(\approx\) \(0.5564543330 - 0.1983301850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.821 - 0.569i)T \)
3 \( 1 + (0.618 - 0.785i)T \)
5 \( 1 + (-0.256 + 0.966i)T \)
7 \( 1 + (0.115 - 0.993i)T \)
11 \( 1 + (-0.942 + 0.335i)T \)
13 \( 1 + (-0.644 + 0.764i)T \)
17 \( 1 + (-0.857 - 0.514i)T \)
19 \( 1 + (-0.0385 + 0.999i)T \)
23 \( 1 + (-0.768 - 0.639i)T \)
29 \( 1 + (0.137 + 0.990i)T \)
31 \( 1 + (0.945 + 0.324i)T \)
37 \( 1 + (0.0715 + 0.997i)T \)
41 \( 1 + (-0.998 + 0.0550i)T \)
43 \( 1 + (-0.381 - 0.924i)T \)
47 \( 1 + (-0.191 - 0.981i)T \)
53 \( 1 + (-0.795 - 0.605i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.480 + 0.876i)T \)
67 \( 1 + (-0.126 + 0.991i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (-0.899 - 0.436i)T \)
79 \( 1 + (-0.968 - 0.250i)T \)
83 \( 1 + (0.731 + 0.681i)T \)
89 \( 1 + (0.224 - 0.974i)T \)
97 \( 1 + (0.999 - 0.0440i)T \)
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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

−23.16121577052946228028934709887, −21.932186289619227192702030729749, −21.22185886256311262490141238818, −20.27609504236120009847475685129, −19.67339741902947393893185820400, −18.95327723169772491303528890616, −17.754696898613251900122931657680, −17.14102158209030168473083458522, −15.90682803557392378839284192030, −15.61296118893995524080122348574, −15.06846392921638040361267154674, −13.80011753178337809399170839050, −12.8971811430406511751637857491, −11.615345408545531454899352839846, −10.71662827613528252387419162868, −9.650277248714215721720466583790, −9.11581425925163422305462824514, −8.03527139662329426864840338812, −7.96549460429323104666793296289, −6.107055651107331877089592272985, −5.197236371741852411795181525005, −4.57587378310068061281472959591, −2.861291130018906689188075198147, −1.96312626098518059719326494762, −0.041446353023392090211414322142, 1.63663127146069848601509496567, 2.50247891270115987893262208648, 3.38570836956383174288715200949, 4.44312407107202048434533255246, 6.58965839743389213148222243621, 7.072243830177477814456906226137, 7.83548339040949861932547183091, 8.6075411023309738640422146450, 9.982287351160749591565587423, 10.37629384929766061078303829907, 11.561287615697468500352082933081, 12.22514186959232653554078701233, 13.35665708029551651513074816747, 14.02089673175648049446695340075, 14.99915379082626627879361356730, 16.077406917158318556133647148228, 17.12029277079098794211145470806, 18.03435541125262150084493923761, 18.51769417981251287996644739312, 19.274209808477031232958828582347, 20.127009409229023159925426277966, 20.57983658079674749981612495189, 21.67910696857477750485234939555, 22.69275460714860170322171703216, 23.57834218875133311279819378047

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