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Properties

Label 1-539-539.304-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.952 - 0.305i$
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  + (−0.0149 − 0.999i)2-s + (0.842 − 0.538i)3-s + (−0.999 + 0.0299i)4-s + (0.163 − 0.986i)5-s + (−0.550 − 0.834i)6-s + (0.0448 + 0.998i)8-s + (0.420 − 0.907i)9-s + (−0.988 − 0.149i)10-s + (−0.826 + 0.563i)12-s + (0.858 − 0.512i)13-s + (−0.393 − 0.919i)15-s + (0.998 − 0.0598i)16-s + (0.971 + 0.237i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + (−0.134 + 0.990i)20-s + ⋯
L(s)  = 1  + (−0.0149 − 0.999i)2-s + (0.842 − 0.538i)3-s + (−0.999 + 0.0299i)4-s + (0.163 − 0.986i)5-s + (−0.550 − 0.834i)6-s + (0.0448 + 0.998i)8-s + (0.420 − 0.907i)9-s + (−0.988 − 0.149i)10-s + (−0.826 + 0.563i)12-s + (0.858 − 0.512i)13-s + (−0.393 − 0.919i)15-s + (0.998 − 0.0598i)16-s + (0.971 + 0.237i)17-s + (−0.913 − 0.406i)18-s + (0.913 − 0.406i)19-s + (−0.134 + 0.990i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2719957413 - 1.735745220i\)
\(L(\frac12)\) \(\approx\) \(0.2719957413 - 1.735745220i\)
\(L(1)\) \(\approx\) \(0.8353373556 - 1.060569077i\)
\(L(1)\) \(\approx\) \(0.8353373556 - 1.060569077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0149 - 0.999i)T \)
3 \( 1 + (0.842 - 0.538i)T \)
5 \( 1 + (0.163 - 0.986i)T \)
13 \( 1 + (0.858 - 0.512i)T \)
17 \( 1 + (0.971 + 0.237i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.983 + 0.178i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.337 + 0.941i)T \)
41 \( 1 + (-0.0448 - 0.998i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.599 + 0.800i)T \)
53 \( 1 + (-0.280 + 0.959i)T \)
59 \( 1 + (-0.887 + 0.460i)T \)
61 \( 1 + (-0.280 - 0.959i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (-0.599 + 0.800i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.858 + 0.512i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
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

−23.816154036991315958940620766040, −22.99189453142562658396457308611, −22.19314462433518827718031461246, −21.465441228874878527772830741609, −20.60298996053901419502455673519, −19.332468741162959082678826907803, −18.65992596694545070580058933837, −18.04571038698396696414915122446, −16.754320736567883122433837213555, −16.08464202871271539595553146836, −15.19475797053243377926209547732, −14.58331455057640844108842955843, −13.81736292775015867145389317800, −13.2681821904061923730631632720, −11.66735835884375915200743092835, −10.503222978116025842254067078374, −9.65314533062288883294509743473, −9.013944295271389586764911939347, −7.74758168995960279599907250378, −7.348522548303526166218542338100, −6.067244238532428863642900904350, −5.222782911593639329118174789752, −3.7737447201822949485810083015, −3.36902454186609988937144185392, −1.7496733057531362316831222756, 0.95044544275585337504015506409, 1.66750566407071246940905359392, 2.96756071847221644352521145212, 3.75184838721813145266250031431, 4.92810577714425809597128299271, 5.95893065673816134495130550419, 7.56734873826583797503197892900, 8.39557184403230870126985234254, 9.069040766208978632593110248650, 9.85693785669321049903550339807, 10.95960191689354689188320210298, 12.214818198777299828592718159991, 12.622970111092831339118145398373, 13.51963688703842603963088214140, 14.09518647761341488724464553117, 15.20711216137556671536132929490, 16.38918729857513043403861114847, 17.41877323718493214813268433111, 18.319442965497961732561487967947, 18.90456187527587768062597841102, 19.98600888028063432737379770508, 20.445719707734987263375604837597, 20.9882742395947254645902077354, 21.96682243591809726446426313459, 23.10236977382681529181178567314

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