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Properties

Label 1-539-539.241-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.896 - 0.443i$
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  + (−0.365 − 0.930i)2-s + (−0.0747 − 0.997i)3-s + (−0.733 + 0.680i)4-s + (−0.826 + 0.563i)5-s + (−0.900 + 0.433i)6-s + (0.900 + 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.826 + 0.563i)10-s + (0.733 + 0.680i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.222 − 0.974i)20-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.0747 − 0.997i)3-s + (−0.733 + 0.680i)4-s + (−0.826 + 0.563i)5-s + (−0.900 + 0.433i)6-s + (0.900 + 0.433i)8-s + (−0.988 + 0.149i)9-s + (0.826 + 0.563i)10-s + (0.733 + 0.680i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (0.222 − 0.974i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.896 - 0.443i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.896 - 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1639114432 - 0.7008609126i\)
\(L(\frac12)\) \(\approx\) \(0.1639114432 - 0.7008609126i\)
\(L(1)\) \(\approx\) \(0.5240994177 - 0.4610051833i\)
\(L(1)\) \(\approx\) \(0.5240994177 - 0.4610051833i\)

Euler product

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

−23.683988967824266425386620703401, −23.19438300647074036572015096572, −22.30132271207261565790618587428, −21.26024162270543342271535941864, −20.477855157572830242760588829822, −19.41070155661112541648954995532, −18.84511941408148941736191668181, −17.52581972973830595243551512065, −16.727669010604956692529488755774, −16.25683071693743863749954122211, −15.40426656910560661802191860267, −14.8222078748013265549056424439, −13.845192938456612867029745833546, −12.724074392480557086725604241251, −11.484245356213086189653800150709, −10.7113518650952113485277791737, −9.61905495701403171683368278025, −8.81169776517223505503457521549, −8.255275581662878809899969284674, −7.06375746785216407247800829978, −6.00652845146830153301048847744, −4.900581571454368975274361505357, −4.35532427810887418122277125982, −3.26399847684296724938422390539, −1.12000179806482170301267974688, 0.54245190840689112293297491368, 1.738063990193276645677475812376, 3.01040105858991130470678116699, 3.58773267167846357064512973155, 5.08436929223775212622096745170, 6.35831344009083216188759835460, 7.56427951309909829892532302222, 8.01995523136379862466523360333, 8.9765082344079958410141393607, 10.40902950786569003075780125213, 10.95162239077407902420095892493, 12.062787137978279793295568082997, 12.358260374090202723724715842671, 13.50216339228396898059070860640, 14.23306024985055382335930305216, 15.36675102006525429266645418627, 16.603021401835485460501690544396, 17.48062742506391060638978075325, 18.321921332305384448269459347620, 18.974590391110960008831270002, 19.434360239111197580578874364284, 20.42163216849405550849315998710, 21.155218752412914266789096141547, 22.45427589972030204837760065221, 23.113134684060838048347359410597

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