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Properties

Label 1-539-539.160-r0-0-0
Degree $1$
Conductor $539$
Sign $0.729 + 0.683i$
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  + (−0.473 + 0.880i)2-s + (0.995 − 0.0896i)3-s + (−0.550 − 0.834i)4-s + (−0.753 + 0.657i)5-s + (−0.393 + 0.919i)6-s + (0.995 − 0.0896i)8-s + (0.983 − 0.178i)9-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)12-s + (0.473 − 0.880i)13-s + (−0.691 + 0.722i)15-s + (−0.393 + 0.919i)16-s + (−0.0448 − 0.998i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.963 + 0.266i)20-s + ⋯
L(s)  = 1  + (−0.473 + 0.880i)2-s + (0.995 − 0.0896i)3-s + (−0.550 − 0.834i)4-s + (−0.753 + 0.657i)5-s + (−0.393 + 0.919i)6-s + (0.995 − 0.0896i)8-s + (0.983 − 0.178i)9-s + (−0.222 − 0.974i)10-s + (−0.623 − 0.781i)12-s + (0.473 − 0.880i)13-s + (−0.691 + 0.722i)15-s + (−0.393 + 0.919i)16-s + (−0.0448 − 0.998i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.963 + 0.266i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.729 + 0.683i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ 0.729 + 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.251094732 + 0.4945509862i\)
\(L(\frac12)\) \(\approx\) \(1.251094732 + 0.4945509862i\)
\(L(1)\) \(\approx\) \(1.008834562 + 0.3537617358i\)
\(L(1)\) \(\approx\) \(1.008834562 + 0.3537617358i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.473 + 0.880i)T \)
3 \( 1 + (0.995 - 0.0896i)T \)
5 \( 1 + (-0.753 + 0.657i)T \)
13 \( 1 + (0.473 - 0.880i)T \)
17 \( 1 + (-0.0448 - 0.998i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.936 + 0.351i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.936 - 0.351i)T \)
41 \( 1 + (-0.995 + 0.0896i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.691 + 0.722i)T \)
53 \( 1 + (-0.0448 + 0.998i)T \)
59 \( 1 + (0.995 + 0.0896i)T \)
61 \( 1 + (-0.0448 - 0.998i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (-0.691 + 0.722i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.473 + 0.880i)T \)
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

−23.45467187664045895147693725180, −22.11827934198935058994524421779, −21.34961905573044486835413917556, −20.662525679590175084825653094993, −19.9395623605859368181838301428, −19.2377922954520843125052207963, −18.76930583328800261119353988571, −17.51947102107272805867807654687, −16.60708488874177798958902673754, −15.72242480729095764094082548585, −14.85049635246090130237318977952, −13.4988884953172161543712900417, −13.174592774123850330939594980508, −12.029007678082819548092843590030, −11.29673945284630264157188979586, −10.21584634305957890331023965661, −9.14085046044103792642769692746, −8.72649256999431311006244588917, −7.84443969429997215226076823061, −6.95920164813253454342179307754, −4.977537385031371968330197618035, −4.02650854642252016264425530036, −3.412428807867471009212299801154, −2.129318071811043975922058044205, −1.12362753371117287674718675309, 0.97384298259619931823419204092, 2.587942266265422005904845499, 3.63227268909514286408213600606, 4.66970945727358530658203644235, 6.02934747812889072678339321326, 7.06725641356126432026085561739, 7.761415773245526062469479290630, 8.3789059877667763316393109500, 9.418446912622300451017867069567, 10.26582959669505993750295221524, 11.20771559574580274741057240378, 12.61353338076229133809922451623, 13.59132245836245361948065775633, 14.44701039488852075620606264129, 15.03847579205585172372339127654, 15.77832933214874200751418411565, 16.50099322549260720164777049426, 17.86344305743943677773564706057, 18.63309923085891080545992647697, 18.98212744111708529287598732592, 20.14463962816879048232053844131, 20.59600271302026126339335089775, 22.15476517263025112515848763876, 22.88952336685913503138052360533, 23.60471814045574116743687891688

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