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Properties

Label 1-539-539.118-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.432 + 0.901i$
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  + (−0.936 + 0.351i)2-s + (−0.473 + 0.880i)3-s + (0.753 − 0.657i)4-s + (0.691 + 0.722i)5-s + (0.134 − 0.990i)6-s + (−0.473 + 0.880i)8-s + (−0.550 − 0.834i)9-s + (−0.900 − 0.433i)10-s + (0.222 + 0.974i)12-s + (0.936 − 0.351i)13-s + (−0.963 + 0.266i)15-s + (0.134 − 0.990i)16-s + (0.858 − 0.512i)17-s + (0.809 + 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.995 + 0.0896i)20-s + ⋯
L(s)  = 1  + (−0.936 + 0.351i)2-s + (−0.473 + 0.880i)3-s + (0.753 − 0.657i)4-s + (0.691 + 0.722i)5-s + (0.134 − 0.990i)6-s + (−0.473 + 0.880i)8-s + (−0.550 − 0.834i)9-s + (−0.900 − 0.433i)10-s + (0.222 + 0.974i)12-s + (0.936 − 0.351i)13-s + (−0.963 + 0.266i)15-s + (0.134 − 0.990i)16-s + (0.858 − 0.512i)17-s + (0.809 + 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.995 + 0.0896i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.432 + 0.901i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.432 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4499224913 + 0.7152597647i\)
\(L(\frac12)\) \(\approx\) \(0.4499224913 + 0.7152597647i\)
\(L(1)\) \(\approx\) \(0.6111846935 + 0.3878343980i\)
\(L(1)\) \(\approx\) \(0.6111846935 + 0.3878343980i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.936 + 0.351i)T \)
3 \( 1 + (-0.473 + 0.880i)T \)
5 \( 1 + (0.691 + 0.722i)T \)
13 \( 1 + (0.936 - 0.351i)T \)
17 \( 1 + (0.858 - 0.512i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.393 - 0.919i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.393 + 0.919i)T \)
41 \( 1 + (0.473 - 0.880i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.963 + 0.266i)T \)
53 \( 1 + (0.858 + 0.512i)T \)
59 \( 1 + (-0.473 - 0.880i)T \)
61 \( 1 + (0.858 - 0.512i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.995 + 0.0896i)T \)
73 \( 1 + (-0.963 + 0.266i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.936 + 0.351i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
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

−23.41030179399723531199272321308, −22.12893053406574820807580310624, −21.30155623822911967537348297862, −20.553292019261557585868123506319, −19.63485532007932603699368100873, −18.82347264498688459095945149269, −18.10634359374806172655340280327, −17.38463202613022971467862097968, −16.64143158336495014305938047708, −16.0868676164220995209377499588, −14.53851877070742392844433933873, −13.36123905921847256731217033882, −12.712146391597784207845935517295, −12.00317741599032648300661320523, −10.93949655228810767382978243626, −10.26131685503582237092148201837, −8.94461694416639004125271564468, −8.47425917637811847875334965933, −7.37768292346692580095428131887, −6.35908103737167906383430418810, −5.68399637442607191305780488763, −4.1720630419845655916884178721, −2.56855018428678034657948942788, −1.67582006646707696483477436021, −0.74818679113603275264929010705, 1.21863745960532518122123775491, 2.65353930835441742368190925512, 3.74964217050990401124433263318, 5.38654577559586692169340480870, 5.937852755297738657053998038008, 6.81985145580856768200743561804, 8.01532677263328382355735733047, 9.07300031048373718413055839519, 9.87803684796714713120255681265, 10.521830580819799457884725246739, 11.18042739260940513059613096054, 12.196357541569957810431997735444, 13.81469490533316282168671056054, 14.56118059267269007703761331874, 15.48404449659608211673376628485, 16.079586597839837257872406187964, 17.1295461612021770919606308568, 17.61476394269803710693383778870, 18.4902662884964634586204236166, 19.24223228457599718328868818012, 20.56395040808915104531553388361, 21.03460034634179786674974788008, 21.9227714300275033808031949026, 23.08520332383595700498569398162, 23.41673791164330013643351365692

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