Properties

Label 2-54-9.5-c6-0-4
Degree $2$
Conductor $54$
Sign $-0.250 + 0.968i$
Analytic cond. $12.4229$
Root an. cond. $3.52461$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.89 − 2.82i)2-s + (15.9 − 27.7i)4-s + (39.5 + 22.8i)5-s + (−245. − 424. i)7-s − 181. i·8-s + 258.·10-s + (873. − 504. i)11-s + (466. − 808. i)13-s + (−2.40e3 − 1.38e3i)14-s + (−512. − 886. i)16-s − 8.09e3i·17-s − 7.72e3·19-s + (1.26e3 − 730. i)20-s + (2.85e3 − 4.94e3i)22-s + (1.18e4 + 6.84e3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.316 + 0.182i)5-s + (−0.714 − 1.23i)7-s − 0.353i·8-s + 0.258·10-s + (0.656 − 0.378i)11-s + (0.212 − 0.368i)13-s + (−0.875 − 0.505i)14-s + (−0.125 − 0.216i)16-s − 1.64i·17-s − 1.12·19-s + (0.158 − 0.0913i)20-s + (0.267 − 0.463i)22-s + (0.973 + 0.562i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $-0.250 + 0.968i$
Analytic conductor: \(12.4229\)
Root analytic conductor: \(3.52461\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :3),\ -0.250 + 0.968i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.38706 - 1.79153i\)
\(L(\frac12)\) \(\approx\) \(1.38706 - 1.79153i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.89 + 2.82i)T \)
3 \( 1 \)
good5 \( 1 + (-39.5 - 22.8i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (245. + 424. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-873. + 504. i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (-466. + 808. i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + 8.09e3iT - 2.41e7T^{2} \)
19 \( 1 + 7.72e3T + 4.70e7T^{2} \)
23 \( 1 + (-1.18e4 - 6.84e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-1.96e3 + 1.13e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (1.70e4 - 2.95e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 9.20e4T + 2.56e9T^{2} \)
41 \( 1 + (-3.10e4 - 1.79e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (-3.45e4 - 5.98e4i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (1.32e4 - 7.62e3i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 - 2.36e5iT - 2.21e10T^{2} \)
59 \( 1 + (-2.21e5 - 1.28e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.99e4 + 3.45e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.60e5 - 2.77e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 4.04e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.93e5T + 1.51e11T^{2} \)
79 \( 1 + (4.49e5 + 7.78e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (1.54e5 - 8.94e4i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + 8.26e5iT - 4.96e11T^{2} \)
97 \( 1 + (3.17e5 + 5.50e5i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−13.68143289047306500716177704260, −12.88630525145678174102640103612, −11.42391979402481627753984866887, −10.42016089154208702987336861924, −9.291707055739653693284856296690, −7.27298058553512326510452311590, −6.15438560297071371165813768455, −4.37689878444401565362220546563, −2.98890384039902888041420914703, −0.820851924587811420712100752356, 2.17032737102348169732812701438, 3.98833667757479110446945930866, 5.72072481346119925741695671204, 6.59893012967823543134982892963, 8.468669117465410446609428694283, 9.503794125947215072596538914525, 11.20083561278081733991352048567, 12.56462219994293186400320034913, 13.04471205880263538564438687407, 14.71421162865111464301597915285

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