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Properties

Label 1-539-539.225-r0-0-0
Degree $1$
Conductor $539$
Sign $0.0183 + 0.999i$
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  + (−0.963 − 0.266i)2-s + (−0.691 − 0.722i)3-s + (0.858 + 0.512i)4-s + (0.983 + 0.178i)5-s + (0.473 + 0.880i)6-s + (−0.691 − 0.722i)8-s + (−0.0448 + 0.998i)9-s + (−0.900 − 0.433i)10-s + (−0.222 − 0.974i)12-s + (−0.963 − 0.266i)13-s + (−0.550 − 0.834i)15-s + (0.473 + 0.880i)16-s + (−0.393 + 0.919i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + (0.753 + 0.657i)20-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.691 − 0.722i)3-s + (0.858 + 0.512i)4-s + (0.983 + 0.178i)5-s + (0.473 + 0.880i)6-s + (−0.691 − 0.722i)8-s + (−0.0448 + 0.998i)9-s + (−0.900 − 0.433i)10-s + (−0.222 − 0.974i)12-s + (−0.963 − 0.266i)13-s + (−0.550 − 0.834i)15-s + (0.473 + 0.880i)16-s + (−0.393 + 0.919i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + (0.753 + 0.657i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0183 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.0183 + 0.999i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.0183 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2597833982 + 0.2645958969i\)
\(L(\frac12)\) \(\approx\) \(0.2597833982 + 0.2645958969i\)
\(L(1)\) \(\approx\) \(0.5326980898 - 0.03888171439i\)
\(L(1)\) \(\approx\) \(0.5326980898 - 0.03888171439i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.963 - 0.266i)T \)
3 \( 1 + (-0.691 - 0.722i)T \)
5 \( 1 + (0.983 + 0.178i)T \)
13 \( 1 + (-0.963 - 0.266i)T \)
17 \( 1 + (-0.393 + 0.919i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.995 - 0.0896i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.995 - 0.0896i)T \)
41 \( 1 + (-0.691 - 0.722i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.550 + 0.834i)T \)
53 \( 1 + (-0.393 - 0.919i)T \)
59 \( 1 + (-0.691 + 0.722i)T \)
61 \( 1 + (-0.393 + 0.919i)T \)
67 \( 1 + T \)
71 \( 1 + (0.753 - 0.657i)T \)
73 \( 1 + (-0.550 - 0.834i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.963 + 0.266i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−23.26823673867083823290040947045, −22.12422830998362575440332950334, −21.65102368624242399668291835092, −20.46866080600910747064948436957, −20.134725866420064253909677015, −18.61137966368178474634668309136, −18.063625085094757435233561536882, −17.1034713250951011826503241625, −16.778026606062158669857084786231, −15.83045095214694475835105245421, −14.94505487438231320669679717849, −14.096455903071125270146988060394, −12.75017882973340885888706213524, −11.65538513982527163336824095873, −10.93674611006258471029828542717, −9.897245170236596614442757974461, −9.49949680976424328627159362518, −8.65479125382455437932580905898, −7.14455191286183883823515323910, −6.47820229440798418783126369633, −5.38579427321299219486243245587, −4.76063926968110059857428420370, −2.944293975094166634739857815, −1.81982642668937617266820465405, −0.27469285854989556456614037522, 1.554363507401654736595659765192, 2.03112031308675924321119392490, 3.40461481263986740502951125933, 5.26461477977891635669180515045, 6.04506967123242851503692846463, 6.995579268855219866419046528478, 7.73108228492165915016007682163, 8.84578510109040851460026399228, 9.944903737738001755911372647493, 10.5040520615122632325861610843, 11.47221662923516387659140535667, 12.40948793339536990139833077214, 13.047060422164662514290633491672, 14.17068090788137600008445400141, 15.304766103180148237649412228, 16.546487337777741055743066147647, 17.14240518991540295300399463793, 17.73644668823612116093510424465, 18.45144227786267230777257714753, 19.2519453870640482814329892677, 20.04934632415957597939284989643, 21.15005904160179311351157404262, 21.93656685995138070243940102741, 22.57932392898230119710637194994, 23.98017317514794041282090673960

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