Properties

Label 2-54-9.2-c6-0-1
Degree $2$
Conductor $54$
Sign $-0.917 + 0.396i$
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 + (−156. + 90.6i)5-s + (104. − 180. i)7-s + 181. i·8-s − 1.02e3·10-s + (−2.30e3 − 1.32e3i)11-s + (−438. − 759. i)13-s + (1.02e3 − 590. i)14-s + (−512. + 886. i)16-s + 4.42e3i·17-s − 4.19e3·19-s + (−5.02e3 − 2.89e3i)20-s + (−7.51e3 − 1.30e4i)22-s + (−1.03e4 + 6.00e3i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.25 + 0.724i)5-s + (0.304 − 0.526i)7-s + 0.353i·8-s − 1.02·10-s + (−1.72 − 0.998i)11-s + (−0.199 − 0.345i)13-s + (0.372 − 0.215i)14-s + (−0.125 + 0.216i)16-s + 0.901i·17-s − 0.611·19-s + (−0.627 − 0.362i)20-s + (−0.706 − 1.22i)22-s + (−0.854 + 0.493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $-0.917 + 0.396i$
Analytic conductor: \(12.4229\)
Root analytic conductor: \(3.52461\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :3),\ -0.917 + 0.396i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0510336 - 0.246590i\)
\(L(\frac12)\) \(\approx\) \(0.0510336 - 0.246590i\)
\(L(4)\) 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 \( 1 + (156. - 90.6i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-104. + 180. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (2.30e3 + 1.32e3i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (438. + 759. i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 - 4.42e3iT - 2.41e7T^{2} \)
19 \( 1 + 4.19e3T + 4.70e7T^{2} \)
23 \( 1 + (1.03e4 - 6.00e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (2.28e3 + 1.32e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (2.90e3 + 5.02e3i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 4.15e4T + 2.56e9T^{2} \)
41 \( 1 + (-6.47e4 + 3.73e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (7.30e4 - 1.26e5i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-2.23e4 - 1.28e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 - 1.97e5iT - 2.21e10T^{2} \)
59 \( 1 + (-3.13e4 + 1.80e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (1.19e4 - 2.07e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.76e5 + 3.05e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 4.96e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.82e5T + 1.51e11T^{2} \)
79 \( 1 + (1.93e5 - 3.34e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (3.59e5 + 2.07e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 4.05e5iT - 4.96e11T^{2} \)
97 \( 1 + (1.78e5 - 3.08e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−14.89540306586370938168099702849, −13.68371976949793278680494821201, −12.60724299463458184754288026029, −11.21724471332990578683538805547, −10.56370129977125782452152545834, −8.095300530039980721331517810146, −7.60651805796803436365391698961, −5.92968073008837305556122524352, −4.25741531740710876721639646924, −2.97269623851501650507693244289, 0.083237511094538350223582725239, 2.38534056675353068575268222684, 4.30529418855622910169977211532, 5.24413076632156456643782165784, 7.34966357915469017915293546236, 8.467196051078018895590559285350, 10.11142616730713505672097448593, 11.50167606444814614684214478260, 12.31453460342391157824175348671, 13.15126343411703491001940063186

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