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Properties

Label 2-539-11.5-c1-0-14
Degree $2$
Conductor $539$
Sign $-0.622 - 0.782i$
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  + (0.596 + 1.83i)2-s + (1.75 + 1.27i)3-s + (−1.39 + 1.01i)4-s + (0.251 − 0.775i)5-s + (−1.29 + 3.97i)6-s + (0.424 + 0.308i)8-s + (0.523 + 1.61i)9-s + 1.57·10-s + (2.84 + 1.70i)11-s − 3.74·12-s + (0.350 + 1.07i)13-s + (1.42 − 1.03i)15-s + (−1.38 + 4.25i)16-s + (1.12 − 3.45i)17-s + (−2.64 + 1.92i)18-s + (−4.27 − 3.10i)19-s + ⋯
L(s)  = 1  + (0.421 + 1.29i)2-s + (1.01 + 0.735i)3-s + (−0.699 + 0.507i)4-s + (0.112 − 0.346i)5-s + (−0.527 + 1.62i)6-s + (0.150 + 0.109i)8-s + (0.174 + 0.537i)9-s + 0.497·10-s + (0.858 + 0.512i)11-s − 1.08·12-s + (0.0970 + 0.298i)13-s + (0.368 − 0.268i)15-s + (−0.345 + 1.06i)16-s + (0.272 − 0.837i)17-s + (−0.623 + 0.453i)18-s + (−0.980 − 0.712i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.622 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13186 + 2.34824i\)
\(L(\frac12)\) \(\approx\) \(1.13186 + 2.34824i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.84 - 1.70i)T \)
good2 \( 1 + (-0.596 - 1.83i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.75 - 1.27i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.251 + 0.775i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.350 - 1.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.12 + 3.45i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.27 + 3.10i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 9.21T + 23T^{2} \)
29 \( 1 + (2.57 - 1.87i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.42 + 4.37i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.29 + 3.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.05 + 5.85i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.12T + 43T^{2} \)
47 \( 1 + (-6.06 - 4.40i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.835 - 2.57i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.88 + 7.18i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.916 - 2.82i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.47T + 67T^{2} \)
71 \( 1 + (-3.15 + 9.72i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.37 - 6.08i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.83 + 5.63i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.14 - 15.8i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.75T + 89T^{2} \)
97 \( 1 + (1.06 + 3.28i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.03703439571660631190433921256, −9.852113968964169765374177521812, −9.121623851250676898662308215668, −8.482754631323950347635137532186, −7.49735358906871835508693366040, −6.63361431170929748952781153619, −5.60097980726194936311568676601, −4.45743952509921742972680011045, −3.88472806510158837586446950268, −2.19402936453009315367458581398, 1.48021636988320059837152799647, 2.36275284147645513914853804138, 3.42882252729113332791986095390, 4.17187251134354550449010764847, 5.90775558927951194164997771801, 6.91919451141275158166173888749, 8.076870410321599667046116856708, 8.665171647710763257053631767154, 9.947589494489359290942298441025, 10.51274264129949133363637554747

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