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Properties

Label 1-539-539.164-r0-0-0
Degree $1$
Conductor $539$
Sign $0.832 + 0.554i$
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  + (0.988 − 0.149i)2-s + (−0.826 − 0.563i)3-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (−0.900 − 0.433i)6-s + (0.900 − 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0747 + 0.997i)10-s + (−0.955 − 0.294i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.222 + 0.974i)20-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (−0.826 − 0.563i)3-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (−0.900 − 0.433i)6-s + (0.900 − 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0747 + 0.997i)10-s + (−0.955 − 0.294i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.222 + 0.974i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.832 + 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.821121065 + 0.5511031405i\)
\(L(\frac12)\) \(\approx\) \(1.821121065 + 0.5511031405i\)
\(L(1)\) \(\approx\) \(1.493279981 + 0.07128437954i\)
\(L(1)\) \(\approx\) \(1.493279981 + 0.07128437954i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.988 - 0.149i)T \)
3 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (0.955 - 0.294i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
97 \( 1 - T \)
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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

−23.396053314078224396035106404033, −22.46914816570405602392231768556, −21.85252750973549380644181851757, −20.87513178015026698538636570934, −20.4554215020705807521881984987, −19.49707952399651271233163825495, −17.90485032982854167963496414791, −17.2246517797949396218494778079, −16.39060502468896579940134863215, −15.56151412140165833331998549933, −15.273775832902966125255968610448, −13.66501612287828721218996734266, −13.13412071523030842209659006668, −12.12557445999249793108065431526, −11.50265693061589942884733317129, −10.63431351764165995052243011702, −9.51737440065436055592214786552, −8.40147067719522903625407666610, −7.24154544827512866607225867167, −6.04804704142198324120459839423, −5.48714678681632865774156400500, −4.466347237109882249413025650445, −3.93501655606736129928344166536, −2.447156986134096466318686041709, −0.833805381458702898785902230823, 1.54005479299361176769624718967, 2.43193353353331935860486688347, 3.7572744002748286136154890444, 4.62602316729805263154157199870, 5.98888811459826626229685602006, 6.41250364774854224835483222447, 7.192305051690820522450719517033, 8.33537010169261355779469413605, 10.21321876237263773110424391074, 10.74931227642129910743019468729, 11.55360698523008307168957787379, 12.295633944092617331960347070672, 13.21199978742349927763284231362, 14.05429757922494988614123843001, 14.7801685055598854284019750829, 15.879818680168951680661683443887, 16.545587545778695062625349476899, 17.67996376082041012630429830371, 18.60781393648263924505181988515, 19.20486073092589402758517293306, 20.1861256493956152715163975351, 21.45955690392970202308504617308, 21.931472479413603255756254341665, 22.72148229547804071442957157138, 23.47452080846377580208745927697

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