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Properties

Label 1-539-539.283-r0-0-0
Degree $1$
Conductor $539$
Sign $0.696 + 0.717i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.696 + 0.717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.586440971 + 1.094192402i\)
\(L(\frac12)\) \(\approx\) \(2.586440971 + 1.094192402i\)
\(L(1)\) \(\approx\) \(1.995455065 + 0.4587333544i\)
\(L(1)\) \(\approx\) \(1.995455065 + 0.4587333544i\)

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

−22.96579811001179508583973614285, −22.6633663665709439814374041546, −21.30803691507627527990669972346, −20.92156239390229179593303346692, −20.24914371368733624816291388811, −19.5176169836529235475622527894, −18.68187748732591245856783818970, −16.990380661182060835022323692079, −16.28705891728743653644276284174, −15.52127158476044170370725813666, −14.804549891790312588572158997060, −13.96644448197458738447791439783, −13.000780775682954529683020601096, −12.31379026339318720748179178854, −11.31468438311499817635807545161, −10.2804173227562613259464909974, −9.657885878787791311300063264229, −8.35135633336472250579419414458, −7.70714355894526488993367397536, −6.02757552927260621514606928070, −5.16815198928085671051802689497, −4.35056562034663912207730606518, −3.52398390675701494374244730462, −2.58439008571851161217366233501, −1.1358595323331573685617379105, 1.65113173568894699191913605411, 2.844518126880004877638429130067, 3.43867981113687745800155036657, 4.5989328072898354786159958439, 6.01141787075555726141769867852, 6.79586166064113618377837379396, 7.430735483468636440954243767956, 8.26254926684490035417144576755, 9.45480562211757288256495858419, 10.95440752701923897490612220479, 11.71378789776029313191990331895, 12.447400689390142392864083770034, 13.57331321250934701986954187254, 14.04723270315966356661630630035, 14.91213042605535512089405468940, 15.5059653370757084468132931191, 16.65958248588333631869390225710, 17.65725015946704991567503563169, 18.63467732900100664266969049966, 19.3862411633022326522296321972, 20.136250503581205758872433108255, 21.256947566936740133597743040007, 21.84203045201847460918204815644, 23.04854759053676009185025088784, 23.488773255724133391834826826643

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