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Properties

Label 1-539-539.170-r0-0-0
Degree $1$
Conductor $539$
Sign $0.805 - 0.591i$
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  + (0.887 − 0.460i)2-s + (0.791 + 0.611i)3-s + (0.575 − 0.817i)4-s + (0.525 − 0.850i)5-s + (0.983 + 0.178i)6-s + (0.134 − 0.990i)8-s + (0.251 + 0.967i)9-s + (0.0747 − 0.997i)10-s + (0.955 − 0.294i)12-s + (−0.0448 + 0.998i)13-s + (0.936 − 0.351i)15-s + (−0.337 − 0.941i)16-s + (0.193 + 0.981i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (−0.393 − 0.919i)20-s + ⋯
L(s)  = 1  + (0.887 − 0.460i)2-s + (0.791 + 0.611i)3-s + (0.575 − 0.817i)4-s + (0.525 − 0.850i)5-s + (0.983 + 0.178i)6-s + (0.134 − 0.990i)8-s + (0.251 + 0.967i)9-s + (0.0747 − 0.997i)10-s + (0.955 − 0.294i)12-s + (−0.0448 + 0.998i)13-s + (0.936 − 0.351i)15-s + (−0.337 − 0.941i)16-s + (0.193 + 0.981i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (−0.393 − 0.919i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.805 - 0.591i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.805 - 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.169090552 - 1.038736462i\)
\(L(\frac12)\) \(\approx\) \(3.169090552 - 1.038736462i\)
\(L(1)\) \(\approx\) \(2.297028117 - 0.5202346361i\)
\(L(1)\) \(\approx\) \(2.297028117 - 0.5202346361i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.887 - 0.460i)T \)
3 \( 1 + (0.791 + 0.611i)T \)
5 \( 1 + (0.525 - 0.850i)T \)
13 \( 1 + (-0.0448 + 0.998i)T \)
17 \( 1 + (0.193 + 0.981i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.0149 - 0.999i)T \)
41 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.163 - 0.986i)T \)
53 \( 1 + (-0.946 + 0.323i)T \)
59 \( 1 + (-0.925 - 0.379i)T \)
61 \( 1 + (-0.946 - 0.323i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.163 + 0.986i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.0448 - 0.998i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
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

−23.44116264484137486038449928778, −22.73525342935031220443713907066, −22.0254044989408847384925525660, −20.96179382803823321050285052729, −20.39271456111084461707160689633, −19.43903657613217356737508501324, −18.15099738420720085061431081001, −17.93841660063064895026619284547, −16.58981668879325241938834393643, −15.525963234234655136403964867949, −14.82907026294294230817951746597, −13.961695014153940586858959306030, −13.652306242794299873490767305905, −12.51649890119491371262506184417, −11.83395956516567277802813160198, −10.53874903398660928830109030731, −9.56440446638285711260385458772, −8.1898815729093455786362357607, −7.60008279457573139691121351856, −6.61881682260035870920039657420, −5.94981241871712855560100444826, −4.7201447309321600026174104174, −3.185840866302802614624605866529, −2.95037960427292157067556515853, −1.661519011070861355906735144327, 1.503956226415219028579445421522, 2.299131984771841093920844245051, 3.55792281939796917944167371636, 4.378129897734977360409051133292, 5.180145937701018153991656628093, 6.16997452602808631976064835556, 7.492773860140334292208681941588, 8.75098069116276403956983347568, 9.52137715917602603358247001977, 10.24558218935997082151916134951, 11.33604044559893344005043118781, 12.32484635251843701621449352060, 13.26342340055461925368229111463, 13.88253112558559262624265339990, 14.58519174470874624281664965753, 15.65154868951736737702324921284, 16.23185010762010140301383566206, 17.23774321060887201628692873424, 18.61941304591861173120624559662, 19.65893539901247028892335921932, 20.06436399684808821074340194815, 21.0009901915630494608938450002, 21.63594582850468887960536567997, 22.02019792979680448161874944436, 23.42528637481051628437383475018

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