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Properties

Label 1-539-539.421-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.784 - 0.619i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.784 - 0.619i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.784 - 0.619i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8256203164 - 2.376779125i\)
\(L(\frac12)\) \(\approx\) \(0.8256203164 - 2.376779125i\)
\(L(1)\) \(\approx\) \(1.381984901 - 1.282333753i\)
\(L(1)\) \(\approx\) \(1.381984901 - 1.282333753i\)

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

−23.36588317970073306052993306678, −23.0446742852142582921844319663, −22.06849781844713209460809716785, −21.34312070175791901277840775590, −20.72582142119772213032694964605, −19.68740686454667364457589895730, −19.03668333059355244492558888262, −17.6875456922163320793014592864, −16.43973966219715321937572854671, −16.04721461939710629586209803002, −15.019437681634422867401803092654, −14.64191642367496744967440406634, −13.77271384554915742399587634482, −12.764865593329473045546177511387, −11.59264528382503313498035658243, −10.945990192754737431870481927710, −10.11528657373313612789887047185, −8.5762344897499814463114342085, −8.03054668193653636777154847003, −6.83375155778943359695120865015, −5.96166628785908555634084414852, −4.70606798214856622779168755311, −3.91818246688104484595874431594, −3.23570144112914393015460636776, −2.192060511679489062854935904954, 0.952462374821991207812604686890, 1.88517603538996852363937783309, 3.28586158483199746350304103244, 3.86398037445610766368293435897, 5.22522552925949537861461841876, 6.06712910484072017531409319774, 7.20616086466786316652882630318, 8.01252020836332158897178897657, 8.9683119777423282197408400127, 10.22273666985504999690376810350, 11.51664236930375502300085811417, 11.98499892930076072073724875410, 12.95587018729239825108121819011, 13.4073384231479763258856834406, 14.44067255565082745821941838746, 15.2388023304523375243117723228, 16.10303942306802788589014675282, 17.07112101848003837515031009140, 18.46320623952680566919930785241, 19.06705455350567700489612271228, 19.90627118939877365002264130683, 20.61419995038691837121422065158, 21.11213889283353800189046396412, 22.493971154068362204232503364081, 23.4218981377063715675654414477

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