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Properties

Label 2-539-11.3-c1-0-3
Degree $2$
Conductor $539$
Sign $-0.868 - 0.495i$
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  + (−1.99 + 1.44i)2-s + (0.5 + 1.53i)3-s + (1.26 − 3.88i)4-s + (−2.80 − 2.03i)5-s + (−3.22 − 2.34i)6-s + (1.58 + 4.89i)8-s + (0.309 − 0.224i)9-s + 8.55·10-s + (2.91 + 1.57i)11-s + 6.60·12-s + (−0.528 + 0.384i)13-s + (1.73 − 5.33i)15-s + (−3.65 − 2.65i)16-s + (−0.919 − 0.668i)17-s + (−0.291 + 0.896i)18-s + (1.87 + 5.77i)19-s + ⋯
L(s)  = 1  + (−1.41 + 1.02i)2-s + (0.288 + 0.888i)3-s + (0.631 − 1.94i)4-s + (−1.25 − 0.911i)5-s + (−1.31 − 0.957i)6-s + (0.561 + 1.72i)8-s + (0.103 − 0.0748i)9-s + 2.70·10-s + (0.880 + 0.474i)11-s + 1.90·12-s + (−0.146 + 0.106i)13-s + (0.447 − 1.37i)15-s + (−0.913 − 0.663i)16-s + (−0.223 − 0.162i)17-s + (−0.0686 + 0.211i)18-s + (0.430 + 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.868 - 0.495i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.868 - 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133565 + 0.504264i\)
\(L(\frac12)\) \(\approx\) \(0.133565 + 0.504264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.91 - 1.57i)T \)
good2 \( 1 + (1.99 - 1.44i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.5 - 1.53i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.80 + 2.03i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (0.528 - 0.384i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.919 + 0.668i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.87 - 5.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.66T + 23T^{2} \)
29 \( 1 + (1.41 - 4.34i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.26 + 1.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.135 - 0.418i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.82 - 5.61i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-0.186 - 0.575i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.94 - 5.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.523 - 1.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.54 - 4.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 + (-4.38 - 3.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.07 - 6.37i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.14 - 1.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.41 - 3.93i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.698T + 89T^{2} \)
97 \( 1 + (12.0 - 8.73i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.80044584032641370351653751315, −9.778457065369599646495532851773, −9.377032808843469192635716019038, −8.504160517451934678935395145068, −7.87039767096912319438953879240, −7.04790936257308529246397009659, −5.86937704438441585562968540504, −4.54024727259950865541387101673, −3.81147481718060232861716148673, −1.26133703331609477442461635164, 0.53953412143106605936152423069, 2.08059041803105408942684592682, 3.10719803641808313271114866772, 4.13136942357903022238739592667, 6.47068419423133800607143305091, 7.32849342696292300473312313888, 7.84653320293808664556728788403, 8.618710764826849181084239634704, 9.583053962708694118007678797810, 10.60121438725313550896164561448

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