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Properties

Label 2-54-9.2-c8-0-6
Degree $2$
Conductor $54$
Sign $-0.444 + 0.895i$
Analytic cond. $21.9984$
Root an. cond. $4.69024$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.79 + 5.65i)2-s + (63.9 + 110. i)4-s + (−683. + 394. i)5-s + (89.7 − 155. i)7-s + 1.44e3i·8-s − 8.92e3·10-s + (4.61e3 + 2.66e3i)11-s + (−2.51e4 − 4.35e4i)13-s + (1.75e3 − 1.01e3i)14-s + (−8.19e3 + 1.41e4i)16-s − 1.16e5i·17-s + 7.72e3·19-s + (−8.74e4 − 5.04e4i)20-s + (3.01e4 + 5.21e4i)22-s + (−3.58e5 + 2.07e5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.09 + 0.631i)5-s + (0.0373 − 0.0647i)7-s + 0.353i·8-s − 0.892·10-s + (0.314 + 0.181i)11-s + (−0.879 − 1.52i)13-s + (0.0457 − 0.0264i)14-s + (−0.125 + 0.216i)16-s − 1.39i·17-s + 0.0592·19-s + (−0.546 − 0.315i)20-s + (0.128 + 0.222i)22-s + (−1.28 + 0.740i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(21.9984\)
Root analytic conductor: \(4.69024\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :4),\ -0.444 + 0.895i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.222162 - 0.358065i\)
\(L(\frac12)\) \(\approx\) \(0.222162 - 0.358065i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.79 - 5.65i)T \)
3 \( 1 \)
good5 \( 1 + (683. - 394. i)T + (1.95e5 - 3.38e5i)T^{2} \)
7 \( 1 + (-89.7 + 155. i)T + (-2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (-4.61e3 - 2.66e3i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + (2.51e4 + 4.35e4i)T + (-4.07e8 + 7.06e8i)T^{2} \)
17 \( 1 + 1.16e5iT - 6.97e9T^{2} \)
19 \( 1 - 7.72e3T + 1.69e10T^{2} \)
23 \( 1 + (3.58e5 - 2.07e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + (3.18e5 + 1.83e5i)T + (2.50e11 + 4.33e11i)T^{2} \)
31 \( 1 + (7.02e5 + 1.21e6i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 - 1.79e6T + 3.51e12T^{2} \)
41 \( 1 + (1.27e6 - 7.35e5i)T + (3.99e12 - 6.91e12i)T^{2} \)
43 \( 1 + (-9.01e5 + 1.56e6i)T + (-5.84e12 - 1.01e13i)T^{2} \)
47 \( 1 + (3.93e6 + 2.27e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + 8.53e6iT - 6.22e13T^{2} \)
59 \( 1 + (1.46e7 - 8.45e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (7.89e6 - 1.36e7i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-1.57e7 - 2.72e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 4.80e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.30e7T + 8.06e14T^{2} \)
79 \( 1 + (-1.73e7 + 3.01e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + (-2.32e6 - 1.34e6i)T + (1.12e15 + 1.95e15i)T^{2} \)
89 \( 1 - 3.46e6iT - 3.93e15T^{2} \)
97 \( 1 + (-1.54e7 + 2.67e7i)T + (-3.91e15 - 6.78e15i)T^{2} \)
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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

−13.33743175462593683685808346980, −12.07528386012163624734462877612, −11.29356520931612394126419140949, −9.802148758071111891274673907509, −7.86366881794643924281158201249, −7.26293511442371323741413434880, −5.56594740640633293980225485928, −4.05496410545442784001119244377, −2.79054448507197729591063731189, −0.11482713464017094499340796144, 1.76711131601169937168228179510, 3.78390187354681791564859734198, 4.69451969799484402523051889627, 6.44966111440585644953492886414, 7.966782824740716459030607404208, 9.269066972807591501461075940282, 10.84096040887647941342448063654, 12.01715640666761170797571782029, 12.50731264305958403241673528424, 14.05679738856145286702793780478

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