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Properties

Label 1-539-539.186-r1-0-0
Degree $1$
Conductor $539$
Sign $-0.640 + 0.768i$
Analytic cond. $57.9235$
Root an. cond. $57.9235$
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  + (−0.955 − 0.294i)2-s + (0.365 − 0.930i)3-s + (0.826 + 0.563i)4-s + (−0.988 + 0.149i)5-s + (−0.623 + 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (0.988 + 0.149i)10-s + (0.826 − 0.563i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + (−0.900 − 0.433i)20-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)2-s + (0.365 − 0.930i)3-s + (0.826 + 0.563i)4-s + (−0.988 + 0.149i)5-s + (−0.623 + 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.733 − 0.680i)9-s + (0.988 + 0.149i)10-s + (0.826 − 0.563i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + (−0.900 − 0.433i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.640 + 0.768i$
Analytic conductor: \(57.9235\)
Root analytic conductor: \(57.9235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (1:\ ),\ -0.640 + 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1405797106 - 0.3000777221i\)
\(L(\frac12)\) \(\approx\) \(-0.1405797106 - 0.3000777221i\)
\(L(1)\) \(\approx\) \(0.5183200588 - 0.2916333998i\)
\(L(1)\) \(\approx\) \(0.5183200588 - 0.2916333998i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.955 - 0.294i)T \)
3 \( 1 + (0.365 - 0.930i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (-0.988 - 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + T \)
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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

−23.691358577928351112196963960, −23.14022323425178003362495369731, −21.96919319622520818530145198462, −20.98268091174428873586729025961, −20.096444863508514521316025393294, −19.74288213878441318377724265726, −18.822520061807308131872787430472, −17.78749351884450981389256593144, −16.79698525806451616766121006688, −16.17137247549604140190251778649, −15.229154405645963551763363049021, −15.0520427434598829946419733772, −13.75601075795773740793082028685, −12.30377103547496981167099373862, −11.393305146675575554649674614022, −10.52941184933345186480291017053, −9.90386960640356727753035054016, −8.7238254160869331632963719707, −8.18301775017733610174863103351, −7.424769177770720027573741973692, −6.00341677224284791238192333816, −5.05516764616655484626766492681, −3.75861896324939122466345031987, −2.940866535150303354456226258263, −1.34111965119109188839170906892, 0.12963970864541172688073086693, 1.05800161110191007822656895768, 2.39597680564646159012912141889, 3.16782849027686887905162550414, 4.42960583133142497357272696013, 6.26893694038955900642233527157, 7.13772904134761947247284542629, 7.62149539538429289061250954397, 8.74220858672707159907621893399, 9.206089684522333065598266427074, 10.66826586863950953805125117210, 11.61190997663295023406034992501, 11.99595066291597760690908728861, 13.05218543931080178130838364270, 14.12378213325176611261637938122, 15.14171585484104876071779017144, 16.08521817555589568686512134429, 16.82533957506426957995142223353, 18.03039225733838151392830223052, 18.53409090387593934981734510883, 19.20879895447672884673940283410, 20.04996536845351999661623796181, 20.47340680737171327184014521125, 21.704363153825998186970107976532, 22.82093370179742507897921741121

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