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Properties

Label 1-539-539.449-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.921 - 0.387i$
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  + (0.753 − 0.657i)2-s + (−0.550 − 0.834i)3-s + (0.134 − 0.990i)4-s + (−0.0448 + 0.998i)5-s + (−0.963 − 0.266i)6-s + (−0.550 − 0.834i)8-s + (−0.393 + 0.919i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)12-s + (0.753 − 0.657i)13-s + (0.858 − 0.512i)15-s + (−0.963 − 0.266i)16-s + (0.473 − 0.880i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (0.983 + 0.178i)20-s + ⋯
L(s)  = 1  + (0.753 − 0.657i)2-s + (−0.550 − 0.834i)3-s + (0.134 − 0.990i)4-s + (−0.0448 + 0.998i)5-s + (−0.963 − 0.266i)6-s + (−0.550 − 0.834i)8-s + (−0.393 + 0.919i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)12-s + (0.753 − 0.657i)13-s + (0.858 − 0.512i)15-s + (−0.963 − 0.266i)16-s + (0.473 − 0.880i)17-s + (0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + (0.983 + 0.178i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.921 - 0.387i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.921 - 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2754936061 - 1.364703116i\)
\(L(\frac12)\) \(\approx\) \(0.2754936061 - 1.364703116i\)
\(L(1)\) \(\approx\) \(0.9051280420 - 0.8033489879i\)
\(L(1)\) \(\approx\) \(0.9051280420 - 0.8033489879i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.753 - 0.657i)T \)
3 \( 1 + (-0.550 - 0.834i)T \)
5 \( 1 + (-0.0448 + 0.998i)T \)
13 \( 1 + (0.753 - 0.657i)T \)
17 \( 1 + (0.473 - 0.880i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.691 - 0.722i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.691 - 0.722i)T \)
41 \( 1 + (-0.550 - 0.834i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.858 + 0.512i)T \)
53 \( 1 + (0.473 + 0.880i)T \)
59 \( 1 + (-0.550 + 0.834i)T \)
61 \( 1 + (0.473 - 0.880i)T \)
67 \( 1 + T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (0.858 - 0.512i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.753 + 0.657i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−23.68450514061879773214549206152, −23.14476140529016670729285889406, −22.0458535172791629322651522861, −21.46684321951073640767067243564, −20.707923779067325660223305590572, −20.08937085590533119651757756815, −18.51173919951701469230621080961, −17.44925643737577591780889721489, −16.68509381088420994850735236224, −16.23699609110222659197791988929, −15.48259093978857702053946942676, −14.517071436928605924959042978378, −13.65661886415687984847712604492, −12.53374917996297549260110378371, −11.99930670397719688348422841252, −11.03493796416438039161299037009, −9.81808853852285240181914575476, −8.80292122223280905665347889179, −8.09870534837539049233916184810, −6.723538631982752224486435885, −5.67947812767177637053277123548, −5.23505290102081912354069893271, −3.95586002514895817401423665416, −3.660094602593327331916230227121, −1.628485374845757291519525702334, 0.611718326169662390924963717216, 2.048466066369542701951537033266, 2.8657285244771068816359428626, 3.96104309865979505739645540689, 5.36406144547531923602500245467, 6.01119526773151009291097214082, 6.95544954295355309674910629349, 7.77642764539113381641497018247, 9.35552024979075692147954816109, 10.52493350677594789099169277584, 11.09035700695458458439308841938, 11.83650222317009087305329705275, 12.71271732356798325818526289622, 13.67728823684068239069922085563, 14.12202220308494023040647612444, 15.28816564122591914626582105369, 16.06817930674105104794814509118, 17.47662933822849585423300456708, 18.38222590101717017610274634620, 18.70211039048426880071299514974, 19.733165998082221851554358590016, 20.50030586134771024244835423576, 21.63179122070068951185813878987, 22.562077603923772253292200003339, 22.74749333853123349809240877269

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