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Properties

Label 2-54-9.2-c8-0-4
Degree $2$
Conductor $54$
Sign $0.459 - 0.888i$
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 + (69.6 − 40.2i)5-s + (370. − 642. i)7-s + 1.44e3i·8-s + 910.·10-s + (1.97e4 + 1.14e4i)11-s + (−6.49e3 − 1.12e4i)13-s + (7.26e3 − 4.19e3i)14-s + (−8.19e3 + 1.41e4i)16-s + 1.05e5i·17-s + 1.18e5·19-s + (8.91e3 + 5.14e3i)20-s + (1.29e5 + 2.23e5i)22-s + (7.74e4 − 4.46e4i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.111 − 0.0643i)5-s + (0.154 − 0.267i)7-s + 0.353i·8-s + 0.0910·10-s + (1.35 + 0.779i)11-s + (−0.227 − 0.394i)13-s + (0.189 − 0.109i)14-s + (−0.125 + 0.216i)16-s + 1.26i·17-s + 0.907·19-s + (0.0557 + 0.0321i)20-s + (0.551 + 0.955i)22-s + (0.276 − 0.159i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.459 - 0.888i$
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: \((2,\ 54,\ (\ :4),\ 0.459 - 0.888i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.56360 + 1.56046i\)
\(L(\frac12)\) \(\approx\) \(2.56360 + 1.56046i\)
\(L(5)\) 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 \( 1 + (-69.6 + 40.2i)T + (1.95e5 - 3.38e5i)T^{2} \)
7 \( 1 + (-370. + 642. i)T + (-2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (-1.97e4 - 1.14e4i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + (6.49e3 + 1.12e4i)T + (-4.07e8 + 7.06e8i)T^{2} \)
17 \( 1 - 1.05e5iT - 6.97e9T^{2} \)
19 \( 1 - 1.18e5T + 1.69e10T^{2} \)
23 \( 1 + (-7.74e4 + 4.46e4i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + (-9.52e5 - 5.49e5i)T + (2.50e11 + 4.33e11i)T^{2} \)
31 \( 1 + (-2.26e5 - 3.92e5i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + 2.12e6T + 3.51e12T^{2} \)
41 \( 1 + (-1.45e6 + 8.38e5i)T + (3.99e12 - 6.91e12i)T^{2} \)
43 \( 1 + (5.69e5 - 9.85e5i)T + (-5.84e12 - 1.01e13i)T^{2} \)
47 \( 1 + (-3.10e6 - 1.79e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + 1.16e7iT - 6.22e13T^{2} \)
59 \( 1 + (5.34e6 - 3.08e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.10e7 + 1.91e7i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (1.84e7 + 3.19e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 1.51e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.91e7T + 8.06e14T^{2} \)
79 \( 1 + (-3.15e7 + 5.46e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + (4.45e7 + 2.57e7i)T + (1.12e15 + 1.95e15i)T^{2} \)
89 \( 1 + 5.17e7iT - 3.93e15T^{2} \)
97 \( 1 + (4.54e6 - 7.87e6i)T + (-3.91e15 - 6.78e15i)T^{2} \)
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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

−14.00736110611996893741773193673, −12.71280352970473626839703935145, −11.83653573279857440049817878267, −10.39717334460800410639438052657, −8.978364908314152289035760111315, −7.50049490773342547731899907041, −6.34097855012261307301507234619, −4.84720861518278828453267032730, −3.51307214900078302874159203174, −1.50620774877006534940278983842, 0.986655029708729568938550980569, 2.72590214765737139995735426059, 4.24082671933152228657824835204, 5.72452320392367424792370139447, 7.00748882678101202252757563401, 8.818725616333571223799512846023, 9.977278478662946726193292456365, 11.57233385915608848098233825602, 11.98719641665910422244821489666, 13.74735295979003908828145220705

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