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Properties

Label 1-539-539.216-r0-0-0
Degree $1$
Conductor $539$
Sign $0.314 + 0.949i$
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  + (0.550 − 0.834i)2-s + (−0.983 − 0.178i)3-s + (−0.393 − 0.919i)4-s + (−0.134 − 0.990i)5-s + (−0.691 + 0.722i)6-s + (−0.983 − 0.178i)8-s + (0.936 + 0.351i)9-s + (−0.900 − 0.433i)10-s + (0.222 + 0.974i)12-s + (−0.550 + 0.834i)13-s + (−0.0448 + 0.998i)15-s + (−0.691 + 0.722i)16-s + (−0.995 − 0.0896i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.858 + 0.512i)20-s + ⋯
L(s)  = 1  + (0.550 − 0.834i)2-s + (−0.983 − 0.178i)3-s + (−0.393 − 0.919i)4-s + (−0.134 − 0.990i)5-s + (−0.691 + 0.722i)6-s + (−0.983 − 0.178i)8-s + (0.936 + 0.351i)9-s + (−0.900 − 0.433i)10-s + (0.222 + 0.974i)12-s + (−0.550 + 0.834i)13-s + (−0.0448 + 0.998i)15-s + (−0.691 + 0.722i)16-s + (−0.995 − 0.0896i)17-s + (0.809 − 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.858 + 0.512i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.314 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04633472473 - 0.03347269601i\)
\(L(\frac12)\) \(\approx\) \(-0.04633472473 - 0.03347269601i\)
\(L(1)\) \(\approx\) \(0.5228006950 - 0.4477679740i\)
\(L(1)\) \(\approx\) \(0.5228006950 - 0.4477679740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.550 - 0.834i)T \)
3 \( 1 + (-0.983 - 0.178i)T \)
5 \( 1 + (-0.134 - 0.990i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (-0.995 - 0.0896i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.753 - 0.657i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.753 + 0.657i)T \)
41 \( 1 + (0.983 + 0.178i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.0448 + 0.998i)T \)
53 \( 1 + (-0.995 + 0.0896i)T \)
59 \( 1 + (-0.983 + 0.178i)T \)
61 \( 1 + (-0.995 - 0.0896i)T \)
67 \( 1 + T \)
71 \( 1 + (0.858 + 0.512i)T \)
73 \( 1 + (-0.0448 + 0.998i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.550 - 0.834i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
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

−23.96766259783746140567498629648, −23.07937843726696956260411376497, −22.57607923865848825389543221496, −21.95518307090817632900956080910, −21.23601534434505397455103475733, −19.96963028007266990035044027947, −18.629204226651688720882899502853, −17.99699055464180665132073528684, −17.293482836970618458682336213056, −16.42807971127637393623830133970, −15.54451749503017590169947822001, −14.9130059262470939652810749781, −14.10567892613922661297505144731, −12.79227730816801773502635255290, −12.35556510662858847142538970937, −11.082865575221754356126268068306, −10.5364722633414045045346463702, −9.28900151870856899960750851073, −7.97753492137837392968916789200, −7.078752356208877481535887572300, −6.36945642795097818644908770876, −5.57915311093959788143236718357, −4.51611309474956280679522593914, −3.64710725052168956810441331953, −2.367926510319354088797988287329, 0.028405949083501563121484969477, 1.3848948925753656341617539020, 2.34518851432904081840043470480, 4.24040742411446832704492212699, 4.49770452957140154131710486162, 5.61991550833895968832591001723, 6.418509471332077539713217661839, 7.70844541938854871160131287179, 9.19524542678446002020439812277, 9.688534419736394162780308741431, 11.16309979939583661977623510421, 11.39875398706686170442220949356, 12.48994978858903752650658900719, 13.00779065887249301496815937892, 13.82268824747187609395799533369, 15.17305998776614150465315306689, 15.9274917667782629323998605577, 17.03195549671856688075145284159, 17.60265022159820474157024179531, 18.76141244198836257707306178702, 19.46844035498726650711872700327, 20.29962745986384198082104484045, 21.32048017912012423141771254659, 21.810841728460874908804620780466, 22.700475641823820546656590271743

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