Properties

Label 2-539-77.53-c1-0-18
Degree $2$
Conductor $539$
Sign $0.999 + 0.0279i$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.49 + 0.530i)2-s + (0.0429 + 0.408i)3-s + (4.12 − 1.83i)4-s + (2.29 − 2.54i)5-s + (−0.323 − 0.996i)6-s + (−5.18 + 3.76i)8-s + (2.76 − 0.588i)9-s + (−4.37 + 7.57i)10-s + (2.88 + 1.63i)11-s + (0.926 + 1.60i)12-s + (−0.672 + 2.06i)13-s + (1.13 + 0.827i)15-s + (4.90 − 5.45i)16-s + (4.41 + 0.939i)17-s + (−6.60 + 2.93i)18-s + (−2.21 − 0.985i)19-s + ⋯
L(s)  = 1  + (−1.76 + 0.375i)2-s + (0.0247 + 0.235i)3-s + (2.06 − 0.917i)4-s + (1.02 − 1.13i)5-s + (−0.132 − 0.406i)6-s + (−1.83 + 1.33i)8-s + (0.923 − 0.196i)9-s + (−1.38 + 2.39i)10-s + (0.870 + 0.492i)11-s + (0.267 + 0.463i)12-s + (−0.186 + 0.573i)13-s + (0.294 + 0.213i)15-s + (1.22 − 1.36i)16-s + (1.07 + 0.227i)17-s + (−1.55 + 0.692i)18-s + (−0.507 − 0.226i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.999 + 0.0279i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.999 + 0.0279i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884500 - 0.0123631i\)
\(L(\frac12)\) \(\approx\) \(0.884500 - 0.0123631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + (-2.88 - 1.63i)T \)
good2 \( 1 + (2.49 - 0.530i)T + (1.82 - 0.813i)T^{2} \)
3 \( 1 + (-0.0429 - 0.408i)T + (-2.93 + 0.623i)T^{2} \)
5 \( 1 + (-2.29 + 2.54i)T + (-0.522 - 4.97i)T^{2} \)
13 \( 1 + (0.672 - 2.06i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.41 - 0.939i)T + (15.5 + 6.91i)T^{2} \)
19 \( 1 + (2.21 + 0.985i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (0.324 + 0.561i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.01 - 0.736i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.37 + 5.97i)T + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (0.524 - 4.98i)T + (-36.1 - 7.69i)T^{2} \)
41 \( 1 + (-2.12 + 1.54i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 + (4.60 + 2.05i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-8.96 - 9.95i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (-7.00 + 3.11i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-9.59 + 10.6i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (3.11 - 5.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.30 - 4.02i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.41 - 2.40i)T + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (9.54 - 2.02i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (2.58 + 7.96i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.38 + 4.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.69 - 8.27i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.12858706337547964479512759386, −9.814836147537479507864036736554, −9.139495587762492676467698498257, −8.491257587830298019558378349078, −7.34715704822330419433813864278, −6.55133879650129842525238774501, −5.54579250134701512506440426272, −4.24519622527247441268891711184, −1.97127697670514129226413736094, −1.16317726802785588795318195639, 1.29104094458087376653051906555, 2.33784215281959216201599086028, 3.48083125050675648561495928183, 5.71424081665158653319528361593, 6.76776819807263760377514976878, 7.25631645985134237771518172565, 8.278682903183948421723383164866, 9.276824150093349301726782839959, 10.08487088783413150755872419169, 10.38930960727939781394105806012

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