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Properties

Label 1-539-539.400-r0-0-0
Degree $1$
Conductor $539$
Sign $0.249 + 0.968i$
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  + (−0.995 + 0.0896i)2-s + (−0.963 + 0.266i)3-s + (0.983 − 0.178i)4-s + (−0.550 − 0.834i)5-s + (0.936 − 0.351i)6-s + (−0.963 + 0.266i)8-s + (0.858 − 0.512i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)12-s + (−0.995 + 0.0896i)13-s + (0.753 + 0.657i)15-s + (0.936 − 0.351i)16-s + (0.134 + 0.990i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.691 − 0.722i)20-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (−0.963 + 0.266i)3-s + (0.983 − 0.178i)4-s + (−0.550 − 0.834i)5-s + (0.936 − 0.351i)6-s + (−0.963 + 0.266i)8-s + (0.858 − 0.512i)9-s + (0.623 + 0.781i)10-s + (−0.900 + 0.433i)12-s + (−0.995 + 0.0896i)13-s + (0.753 + 0.657i)15-s + (0.936 − 0.351i)16-s + (0.134 + 0.990i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + (−0.691 − 0.722i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2319851664 + 0.1798366557i\)
\(L(\frac12)\) \(\approx\) \(0.2319851664 + 0.1798366557i\)
\(L(1)\) \(\approx\) \(0.4085695636 + 0.01766595652i\)
\(L(1)\) \(\approx\) \(0.4085695636 + 0.01766595652i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.995 + 0.0896i)T \)
3 \( 1 + (-0.963 + 0.266i)T \)
5 \( 1 + (-0.550 - 0.834i)T \)
13 \( 1 + (-0.995 + 0.0896i)T \)
17 \( 1 + (0.134 + 0.990i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.473 - 0.880i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.753 - 0.657i)T \)
53 \( 1 + (0.134 - 0.990i)T \)
59 \( 1 + (-0.963 - 0.266i)T \)
61 \( 1 + (0.134 + 0.990i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (0.753 + 0.657i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.995 - 0.0896i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
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

−23.39255736907197425703433981772, −22.28558174699326953423840022036, −21.79578694307449234791068994917, −20.54275953886511755498869842422, −19.614037253464374677842547745679, −18.77835772247364798589316985786, −18.342386924023116515906559128716, −17.388504703031933749444736149842, −16.72633137764180430246614229456, −15.79675050601238804847302494325, −15.10108004037742834626065722095, −13.92189582620162395989175924684, −12.35319448253090080608023619445, −11.96249478818876584425767898084, −11.0408587652647891939490310431, −10.32706471930536520383146179189, −9.57518467242326552811249750030, −8.08090205291437103229524722525, −7.39509123948173630011123906220, −6.66630424136949950837179554085, −5.74307557241816990998989253292, −4.357330795153277109902489703156, −2.95745657223959415548386483403, −1.88513509090489729929636250546, −0.31981461865366712714137688957, 0.892355659355009985694463477649, 2.19618027904776817855033575802, 3.926324085925219852992363708757, 4.90148123374918556828491425442, 5.96167079259922739351256236107, 6.87507036547063325171882592566, 7.90626634319278921500361293956, 8.755934198458370768450418498957, 9.79023003447695453750250013719, 10.49679771832384504011415677508, 11.48765878921480444598461933572, 12.19713672109342326076672327727, 12.850221111995153772711489103193, 14.64188958972766543341392131134, 15.52971910061531292342450631606, 16.14939207873037113024797583027, 17.11218626939608477741224647704, 17.323853169372692682508136505897, 18.45716409723485036488151950523, 19.4696163808943284897344100787, 19.95410106741019011356822069700, 21.187958463038602975005390121320, 21.64492670393633698100534529274, 22.968003704678165201554954296923, 23.84785095550059123154309711256

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