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Properties

Label 2-54-9.4-c5-0-0
Degree $2$
Conductor $54$
Sign $-0.963 + 0.268i$
Analytic cond. $8.66072$
Root an. cond. $2.94291$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−7.99 + 13.8i)4-s + (−33.0 + 57.2i)5-s + (−57.0 − 98.8i)7-s − 63.9·8-s − 264.·10-s + (−192. − 333. i)11-s + (−516. + 894. i)13-s + (228. − 395. i)14-s + (−128 − 221. i)16-s − 959.·17-s − 464.·19-s + (−528. − 915. i)20-s + (771. − 1.33e3i)22-s + (1.15e3 − 1.99e3i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.591 + 1.02i)5-s + (−0.440 − 0.762i)7-s − 0.353·8-s − 0.835·10-s + (−0.480 − 0.832i)11-s + (−0.847 + 1.46i)13-s + (0.311 − 0.539i)14-s + (−0.125 − 0.216i)16-s − 0.804·17-s − 0.295·19-s + (−0.295 − 0.511i)20-s + (0.339 − 0.588i)22-s + (0.453 − 0.786i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $-0.963 + 0.268i$
Analytic conductor: \(8.66072\)
Root analytic conductor: \(2.94291\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :5/2),\ -0.963 + 0.268i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0894163 - 0.653415i\)
\(L(\frac12)\) \(\approx\) \(0.0894163 - 0.653415i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
3 \( 1 \)
good5 \( 1 + (33.0 - 57.2i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (57.0 + 98.8i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (192. + 333. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (516. - 894. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 959.T + 1.41e6T^{2} \)
19 \( 1 + 464.T + 2.47e6T^{2} \)
23 \( 1 + (-1.15e3 + 1.99e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-3.54e3 - 6.14e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (3.88e3 - 6.72e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 9.31e3T + 6.93e7T^{2} \)
41 \( 1 + (6.66e3 - 1.15e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.05e3 - 1.82e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.24e3 - 2.16e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 1.00e4T + 4.18e8T^{2} \)
59 \( 1 + (-2.72e3 + 4.71e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.70e4 + 2.96e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.67e4 - 4.64e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 970.T + 1.80e9T^{2} \)
73 \( 1 + 7.24e4T + 2.07e9T^{2} \)
79 \( 1 + (1.60e4 + 2.78e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (1.80e4 + 3.12e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + 4.26e4T + 5.58e9T^{2} \)
97 \( 1 + (-2.19e4 - 3.79e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−14.78837241739952307628629729423, −14.10656986360518281243995135463, −12.92658965805635315478469782443, −11.47398031446420954646124767010, −10.46044731055176331369633442220, −8.790422168612297218157060812261, −7.22292639764494092591189706509, −6.59078296955002297522495643640, −4.53490176504914242993493849307, −3.07735123815823633755272702190, 0.27792661700623523939983886123, 2.53224255305849468084600893094, 4.43406230000400061895133923242, 5.61873195189699479313680989217, 7.73114282791695577115830882026, 9.065953807690139203225396547479, 10.20382135677071381570078659199, 11.74704947182768187684363765687, 12.61732919423252898629563147953, 13.21671702604079527701742686986

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