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Properties

Label 2-54-9.2-c8-0-7
Degree $2$
Conductor $54$
Sign $0.754 + 0.656i$
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 + (719. − 415. i)5-s + (1.34e3 − 2.32e3i)7-s + 1.44e3i·8-s + 9.39e3·10-s + (−2.13e4 − 1.23e4i)11-s + (−4.84e3 − 8.39e3i)13-s + (2.63e4 − 1.51e4i)14-s + (−8.19e3 + 1.41e4i)16-s − 4.26e4i·17-s + 1.01e5·19-s + (9.20e4 + 5.31e4i)20-s + (−1.39e5 − 2.41e5i)22-s + (3.96e5 − 2.29e5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.15 − 0.664i)5-s + (0.559 − 0.969i)7-s + 0.353i·8-s + 0.939·10-s + (−1.45 − 0.842i)11-s + (−0.169 − 0.294i)13-s + (0.685 − 0.395i)14-s + (−0.125 + 0.216i)16-s − 0.510i·17-s + 0.776·19-s + (0.575 + 0.332i)20-s + (−0.595 − 1.03i)22-s + (1.41 − 0.819i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.754 + 0.656i$
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: \((2,\ 54,\ (\ :4),\ 0.754 + 0.656i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.01642 - 1.12870i\)
\(L(\frac12)\) \(\approx\) \(3.01642 - 1.12870i\)
\(L(5)\) 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 \( 1 + (-719. + 415. i)T + (1.95e5 - 3.38e5i)T^{2} \)
7 \( 1 + (-1.34e3 + 2.32e3i)T + (-2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (2.13e4 + 1.23e4i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + (4.84e3 + 8.39e3i)T + (-4.07e8 + 7.06e8i)T^{2} \)
17 \( 1 + 4.26e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.01e5T + 1.69e10T^{2} \)
23 \( 1 + (-3.96e5 + 2.29e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + (5.77e5 + 3.33e5i)T + (2.50e11 + 4.33e11i)T^{2} \)
31 \( 1 + (-7.09e5 - 1.22e6i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 - 1.62e6T + 3.51e12T^{2} \)
41 \( 1 + (1.99e6 - 1.15e6i)T + (3.99e12 - 6.91e12i)T^{2} \)
43 \( 1 + (-8.12e5 + 1.40e6i)T + (-5.84e12 - 1.01e13i)T^{2} \)
47 \( 1 + (-3.38e6 - 1.95e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + 6.96e6iT - 6.22e13T^{2} \)
59 \( 1 + (5.94e6 - 3.43e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.26e6 + 2.19e6i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-1.58e7 - 2.74e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 1.75e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.86e7T + 8.06e14T^{2} \)
79 \( 1 + (3.40e7 - 5.89e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + (4.73e7 + 2.73e7i)T + (1.12e15 + 1.95e15i)T^{2} \)
89 \( 1 - 7.04e6iT - 3.93e15T^{2} \)
97 \( 1 + (-5.14e7 + 8.90e7i)T + (-3.91e15 - 6.78e15i)T^{2} \)
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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

−13.49679499652041941681576420332, −12.88479840138768157177349098760, −11.17474370657635960331854789148, −10.07875565105555132091868788644, −8.524858587154768753514844764956, −7.27072837138666488057228532251, −5.61831324591153852023572183178, −4.81143947532103271192421168828, −2.80941538971925220039871786973, −0.946461126396598787267795307644, 1.87670691091447628561302540890, 2.77931110449134155819151645680, 5.00546735566145990168775543498, 5.88973085127238475395609595581, 7.48013760926215557061714983954, 9.340221209621029243296073016408, 10.36258177087160878015181047433, 11.48164281985268626661427857718, 12.79890976749064487084459132320, 13.65297970088401341226153701786

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