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Properties

Label 2-54-9.5-c2-0-0
Degree $2$
Conductor $54$
Sign $0.703 - 0.710i$
Analytic cond. $1.47139$
Root an. cond. $1.21301$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (4.5 + 2.59i)5-s + (4.17 + 7.22i)7-s + 2.82i·8-s − 7.34·10-s + (−0.825 + 0.476i)11-s + (4.84 − 8.39i)13-s + (−10.2 − 5.90i)14-s + (−2.00 − 3.46i)16-s − 18.8i·17-s − 24.6·19-s + (8.99 − 5.19i)20-s + (0.674 − 1.16i)22-s + (−0.825 − 0.476i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.900 + 0.519i)5-s + (0.596 + 1.03i)7-s + 0.353i·8-s − 0.734·10-s + (−0.0750 + 0.0433i)11-s + (0.372 − 0.645i)13-s + (−0.730 − 0.421i)14-s + (−0.125 − 0.216i)16-s − 1.11i·17-s − 1.29·19-s + (0.449 − 0.259i)20-s + (0.0306 − 0.0530i)22-s + (−0.0359 − 0.0207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(1.47139\)
Root analytic conductor: \(1.21301\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :1),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.935931 + 0.390207i\)
\(L(\frac12)\) \(\approx\) \(0.935931 + 0.390207i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-4.5 - 2.59i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.17 - 7.22i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.825 - 0.476i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.84 + 8.39i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 18.8iT - 289T^{2} \)
19 \( 1 + 24.6T + 361T^{2} \)
23 \( 1 + (0.825 + 0.476i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (11.8 - 6.84i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (1.52 - 2.63i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 46.6T + 1.36e3T^{2} \)
41 \( 1 + (-9.45 - 5.45i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.5 + 39.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (39.2 - 22.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 94.3iT - 2.80e3T^{2} \)
59 \( 1 + (-16.2 - 9.39i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (6.54 + 11.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (37.5 - 64.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 18.0iT - 5.04e3T^{2} \)
73 \( 1 + 7.90T + 5.32e3T^{2} \)
79 \( 1 + (-21.8 - 37.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-112. + 65.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-54.9 - 95.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−15.19424444203291778637267969725, −14.44690765156459786867907858091, −13.11915203262088770438518397429, −11.58493321099851535231019597427, −10.43669499780677752900430125500, −9.268836324445037925403635260131, −8.124185686916382445432521365774, −6.49664812410755589559403732794, −5.33892658796957644741213937444, −2.33306185666327441288755425339, 1.65153521327229550390934101392, 4.27572217217527577589462190887, 6.25659367932188497082465774961, 7.896440313459928808872460266923, 9.093540035510732274424143048834, 10.30932528085668470810545272604, 11.21624057442467680263613425235, 12.79672110576093955537705909403, 13.65281874160969723183124765667, 14.91651964856648928465677083397

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