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Properties

Label 2-540-180.59-c1-0-24
Degree $2$
Conductor $540$
Sign $-0.986 + 0.162i$
Analytic cond. $4.31192$
Root an. cond. $2.07651$
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  + (−1.00 − 0.992i)2-s + (0.0291 + 1.99i)4-s + (0.493 − 2.18i)5-s + (−1.65 − 2.87i)7-s + (1.95 − 2.04i)8-s + (−2.66 + 1.70i)10-s + (−1.17 − 2.04i)11-s + (3.81 + 2.20i)13-s + (−1.18 + 4.53i)14-s + (−3.99 + 0.116i)16-s − 0.889·17-s + 3.03i·19-s + (4.37 + 0.923i)20-s + (−0.839 + 3.22i)22-s + (−6.29 − 3.63i)23-s + ⋯
L(s)  = 1  + (−0.712 − 0.701i)2-s + (0.0145 + 0.999i)4-s + (0.220 − 0.975i)5-s + (−0.626 − 1.08i)7-s + (0.691 − 0.722i)8-s + (−0.841 + 0.539i)10-s + (−0.355 − 0.615i)11-s + (1.05 + 0.610i)13-s + (−0.315 + 1.21i)14-s + (−0.999 + 0.0291i)16-s − 0.215·17-s + 0.696i·19-s + (0.978 + 0.206i)20-s + (−0.179 + 0.688i)22-s + (−1.31 − 0.758i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(540\)    =    \(2^{2} \cdot 3^{3} \cdot 5\)
Sign: $-0.986 + 0.162i$
Analytic conductor: \(4.31192\)
Root analytic conductor: \(2.07651\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{540} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 540,\ (\ :1/2),\ -0.986 + 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0562198 - 0.688619i\)
\(L(\frac12)\) \(\approx\) \(0.0562198 - 0.688619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.00 + 0.992i)T \)
3 \( 1 \)
5 \( 1 + (-0.493 + 2.18i)T \)
good7 \( 1 + (1.65 + 2.87i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.17 + 2.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.81 - 2.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.889T + 17T^{2} \)
19 \( 1 - 3.03iT - 19T^{2} \)
23 \( 1 + (6.29 + 3.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.03 - 0.598i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.205 - 0.118i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.4iT - 37T^{2} \)
41 \( 1 + (2.45 + 1.41i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.17 + 7.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.55 - 3.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.772T + 53T^{2} \)
59 \( 1 + (1.06 - 1.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.91 - 3.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.17 - 5.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 6.10iT - 73T^{2} \)
79 \( 1 + (-8.94 + 5.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.25 - 1.29i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.68iT - 89T^{2} \)
97 \( 1 + (5.34 - 3.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35879490887463520447478782875, −9.603274208137437138356342686321, −8.678484171935144466703395308428, −8.074721932123125595435367127710, −6.92735120363240719300812532033, −5.84233565963134730065211303012, −4.24086162567241156983817387477, −3.61324784511725479291141749251, −1.87282540967078559726520008846, −0.49540755558242540853056751895, 2.01297458781553754904633079697, 3.24960696304719735425211292003, 5.05271333859201763869464034106, 6.12256273551131288108661485900, 6.53289067780906607954050980789, 7.72736401046402833978221852334, 8.490774074153183931657763747762, 9.597265679267403302366117164286, 10.03486173749567594169073510520, 11.04633549064368505068532120305

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