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Properties

Label 1-539-539.456-r0-0-0
Degree $1$
Conductor $539$
Sign $0.894 - 0.446i$
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  + (0.983 − 0.178i)2-s + (0.858 − 0.512i)3-s + (0.936 − 0.351i)4-s + (−0.393 + 0.919i)5-s + (0.753 − 0.657i)6-s + (0.858 − 0.512i)8-s + (0.473 − 0.880i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (0.983 − 0.178i)13-s + (0.134 + 0.990i)15-s + (0.753 − 0.657i)16-s + (−0.963 + 0.266i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + (−0.0448 + 0.998i)20-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)2-s + (0.858 − 0.512i)3-s + (0.936 − 0.351i)4-s + (−0.393 + 0.919i)5-s + (0.753 − 0.657i)6-s + (0.858 − 0.512i)8-s + (0.473 − 0.880i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (0.983 − 0.178i)13-s + (0.134 + 0.990i)15-s + (0.753 − 0.657i)16-s + (−0.963 + 0.266i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + (−0.0448 + 0.998i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (456, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.894 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.270166136 - 0.7700033631i\)
\(L(\frac12)\) \(\approx\) \(3.270166136 - 0.7700033631i\)
\(L(1)\) \(\approx\) \(2.343463323 - 0.4044903959i\)
\(L(1)\) \(\approx\) \(2.343463323 - 0.4044903959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.983 - 0.178i)T \)
3 \( 1 + (0.858 - 0.512i)T \)
5 \( 1 + (-0.393 + 0.919i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (-0.963 + 0.266i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.550 - 0.834i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.550 - 0.834i)T \)
41 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.134 - 0.990i)T \)
53 \( 1 + (-0.963 - 0.266i)T \)
59 \( 1 + (0.858 + 0.512i)T \)
61 \( 1 + (-0.963 + 0.266i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.0448 - 0.998i)T \)
73 \( 1 + (0.134 + 0.990i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.983 + 0.178i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−23.66906490961002035297100861903, −22.50295678772568374474960983394, −21.83620339468649996995400986406, −20.80250867146999024670182862902, −20.42736232605345662117099058780, −19.759273240389086568596925996704, −18.70113927144539710395772329779, −17.2262081400189003527313082642, −16.2664546919351353269545877840, −15.76405580998811335127713077297, −15.05359392242353649521552807736, −14.043069215842691499427566365314, −13.23977660469012750635544887186, −12.73755685895633327653295225519, −11.40126034120576566097852590429, −10.80192003283885514689348809576, −9.26306867032734562818401107764, −8.63876243725216206681276899612, −7.66602031652713009551257417184, −6.6647806336157855698039662567, −5.28457515793855786674683132734, −4.551983920833514451360162081449, −3.78163410581203449841087849308, −2.77332517076843315886139810365, −1.55492489904953081586491856881, 1.5069108381655462163305238120, 2.49549259881936386937352812074, 3.53061617740631359106803663566, 3.9865566222208993227792118169, 5.65515981912125700891509054927, 6.57456245515934307389999777139, 7.333795698968534733118702070823, 8.199084540263170800598154570756, 9.48772853445037327916354247741, 10.67447967944629550391247440019, 11.35698795591464146634431110195, 12.38543229275463962837891469539, 13.25906918205342142536284753765, 13.93402181732114273669596546047, 14.7347039976643334502706487495, 15.39339744665616406409283512456, 16.12770596363146559908083612827, 17.716470854394477588769314846096, 18.61026848603967474570650684628, 19.339310647249334901529443553963, 20.028671795261906904949401154642, 20.910036719430707740373687610264, 21.621552882341357235701366257337, 22.76471583881769830086501826166, 23.205428598715397612125343579182

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