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Properties

Label 1-539-539.256-r0-0-0
Degree $1$
Conductor $539$
Sign $0.852 - 0.522i$
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  + (−0.925 − 0.379i)2-s + (−0.599 + 0.800i)3-s + (0.712 + 0.701i)4-s + (0.420 − 0.907i)5-s + (0.858 − 0.512i)6-s + (−0.393 − 0.919i)8-s + (−0.280 − 0.959i)9-s + (−0.733 + 0.680i)10-s + (−0.988 + 0.149i)12-s + (0.134 + 0.990i)13-s + (0.473 + 0.880i)15-s + (0.0149 + 0.999i)16-s + (0.998 + 0.0598i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.936 − 0.351i)20-s + ⋯
L(s)  = 1  + (−0.925 − 0.379i)2-s + (−0.599 + 0.800i)3-s + (0.712 + 0.701i)4-s + (0.420 − 0.907i)5-s + (0.858 − 0.512i)6-s + (−0.393 − 0.919i)8-s + (−0.280 − 0.959i)9-s + (−0.733 + 0.680i)10-s + (−0.988 + 0.149i)12-s + (0.134 + 0.990i)13-s + (0.473 + 0.880i)15-s + (0.0149 + 0.999i)16-s + (0.998 + 0.0598i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + (0.936 − 0.351i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7232914436 - 0.2038843146i\)
\(L(\frac12)\) \(\approx\) \(0.7232914436 - 0.2038843146i\)
\(L(1)\) \(\approx\) \(0.6585339610 - 0.06418232919i\)
\(L(1)\) \(\approx\) \(0.6585339610 - 0.06418232919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.925 - 0.379i)T \)
3 \( 1 + (-0.599 + 0.800i)T \)
5 \( 1 + (0.420 - 0.907i)T \)
13 \( 1 + (0.134 + 0.990i)T \)
17 \( 1 + (0.998 + 0.0598i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.0448 - 0.998i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.887 + 0.460i)T \)
41 \( 1 + (-0.393 - 0.919i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.525 + 0.850i)T \)
53 \( 1 + (-0.447 + 0.894i)T \)
59 \( 1 + (0.992 - 0.119i)T \)
61 \( 1 + (-0.447 - 0.894i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.936 + 0.351i)T \)
73 \( 1 + (0.525 - 0.850i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.134 - 0.990i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
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

−23.43015457283009040152789888357, −22.97220461287892476151557714976, −22.00926785606007102538747034477, −20.83106488663558987861677650066, −19.83670913439104328849140893596, −18.922097517461970130777747114906, −18.268549071161308872965803335626, −17.914164107390386141924119000332, −16.79243724777534313384590186545, −16.312783209847932540335872426417, −14.84739379192330663716180172162, −14.44608954685328614047055405305, −13.172561510525991149913753890580, −12.204383981764708269455334172243, −11.128732062226837139731658832011, −10.5176529326123210660371595921, −9.752579429507337242547353725067, −8.33807277185689667110217427877, −7.61396166218938147887324529157, −6.79639396947187865782980696356, −5.94868159254667490816115346559, −5.30998797658028033422605995044, −3.17781803344708868849774930841, −2.117755570746039505247097454396, −1.01115984503037288978047148373, 0.74374483640730315498258985607, 1.94108389193928020845497202734, 3.41189201463285300086565220986, 4.435255081075147975268516610294, 5.51659618101981055825893532404, 6.503662435914715003440253906093, 7.70394199326191147267696688659, 8.99024886105403724842222827905, 9.32497673853158847931793077725, 10.20144169993909373375505319466, 11.211608853354128024765715477415, 11.88867399940995618047076561569, 12.735444262958856694377829855971, 13.88616642492862502853044184025, 15.29717870434715139576132807832, 15.99296084596707556044382957040, 16.95811032471661991885434378965, 17.105151240970971822905370537326, 18.18005614485012232450626372768, 19.156623891762449337002759377836, 20.13656047948253343012652684496, 20.89378407247074417362135463744, 21.46946047158339858072061058704, 22.093937388314212032928749247443, 23.53035782821215720844730609340

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