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Properties

Label 1-539-539.498-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.817 + 0.575i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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  + (0.858 + 0.512i)2-s + (−0.0448 + 0.998i)3-s + (0.473 + 0.880i)4-s + (0.936 + 0.351i)5-s + (−0.550 + 0.834i)6-s + (−0.0448 + 0.998i)8-s + (−0.995 − 0.0896i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)12-s + (0.858 + 0.512i)13-s + (−0.393 + 0.919i)15-s + (−0.550 + 0.834i)16-s + (−0.691 − 0.722i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.134 + 0.990i)20-s + ⋯
L(s)  = 1  + (0.858 + 0.512i)2-s + (−0.0448 + 0.998i)3-s + (0.473 + 0.880i)4-s + (0.936 + 0.351i)5-s + (−0.550 + 0.834i)6-s + (−0.0448 + 0.998i)8-s + (−0.995 − 0.0896i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)12-s + (0.858 + 0.512i)13-s + (−0.393 + 0.919i)15-s + (−0.550 + 0.834i)16-s + (−0.691 − 0.722i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.134 + 0.990i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (498, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7507279243 + 2.369602730i\)
\(L(\frac12)\) \(\approx\) \(0.7507279243 + 2.369602730i\)
\(L(1)\) \(\approx\) \(1.289533830 + 1.320691368i\)
\(L(1)\) \(\approx\) \(1.289533830 + 1.320691368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.858 + 0.512i)T \)
3 \( 1 + (-0.0448 + 0.998i)T \)
5 \( 1 + (0.936 + 0.351i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (-0.691 - 0.722i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.983 + 0.178i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.983 + 0.178i)T \)
41 \( 1 + (-0.0448 + 0.998i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (-0.393 - 0.919i)T \)
53 \( 1 + (-0.691 + 0.722i)T \)
59 \( 1 + (-0.0448 - 0.998i)T \)
61 \( 1 + (-0.691 - 0.722i)T \)
67 \( 1 + T \)
71 \( 1 + (0.134 - 0.990i)T \)
73 \( 1 + (-0.393 + 0.919i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.858 - 0.512i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−23.17630093852984536402601893767, −22.20926033684649746858594355095, −21.48339365874933669157467718954, −20.609758595871881013347960834211, −19.80320435886764618875740447942, −19.117745783740874739392840064572, −17.944903412776342917857149149208, −17.543590415986934725960925297155, −16.23539402782885946510386405885, −15.21836395378209013025900445350, −14.100302284783217315078569302980, −13.56960081541112991719580137710, −12.850381054678815449637347612718, −12.23999015153138274005841535784, −11.06659513260838010154921645749, −10.3978591757899334930347609593, −9.11605757223538637527946635306, −8.19057843757270280866571961852, −6.714373931843875891869793703, −6.14872970990203190495019536343, −5.35934734070347706719368938205, −4.14831244516586054474393829735, −2.78214130497983271234502105574, −1.960310806390361109575771602647, −1.00933842596566934447958338696, 2.10619813932429202653136425766, 3.05624620037258996325263453492, 4.18028215798576108187628154086, 4.88742099830916532719766494645, 6.215885090270705097822744380238, 6.31894153495995665649095047267, 7.99184844176693667473640447774, 8.93206418469080475294281815373, 9.90521062090798132715015555715, 10.916221795031881874894239312515, 11.608240903303298283229840970412, 12.87320946885653286507177813946, 13.82512115572273472571635320937, 14.338943294530097416781150110082, 15.21938489691059690953262540796, 16.09385902083046900271437025218, 16.72651455286401062026269802282, 17.603791042821929507384255396772, 18.47737340200751490999675214289, 20.04504095637750142745807372450, 20.76362011229632539749250870232, 21.48793222120736656587930724030, 21.99246424838794917266903666931, 22.85494608302820288240450602087, 23.49079003914174892568422582557

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