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Properties

Label 2-54-9.2-c4-0-2
Degree $2$
Conductor $54$
Sign $0.940 - 0.339i$
Analytic cond. $5.58197$
Root an. cond. $2.36262$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 + 1.41i)2-s + (3.99 + 6.92i)4-s + (32.5 − 18.7i)5-s + (−1.35 + 2.34i)7-s + 22.6i·8-s + 106.·10-s + (91.4 + 52.8i)11-s + (97.9 + 169. i)13-s + (−6.63 + 3.83i)14-s + (−32.0 + 55.4i)16-s − 448. i·17-s − 501.·19-s + (260. + 150. i)20-s + (149. + 258. i)22-s + (−343. + 198. i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.30 − 0.750i)5-s + (−0.0276 + 0.0478i)7-s + 0.353i·8-s + 1.06·10-s + (0.756 + 0.436i)11-s + (0.579 + 1.00i)13-s + (−0.0338 + 0.0195i)14-s + (−0.125 + 0.216i)16-s − 1.55i·17-s − 1.39·19-s + (0.650 + 0.375i)20-s + (0.308 + 0.534i)22-s + (−0.648 + 0.374i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(5.58197\)
Root analytic conductor: \(2.36262\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :2),\ 0.940 - 0.339i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.51866 + 0.440241i\)
\(L(\frac12)\) \(\approx\) \(2.51866 + 0.440241i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.44 - 1.41i)T \)
3 \( 1 \)
good5 \( 1 + (-32.5 + 18.7i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (1.35 - 2.34i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-91.4 - 52.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-97.9 - 169. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 448. iT - 8.35e4T^{2} \)
19 \( 1 + 501.T + 1.30e5T^{2} \)
23 \( 1 + (343. - 198. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (466. + 269. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (297. + 515. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.84e3T + 1.87e6T^{2} \)
41 \( 1 + (-145. + 83.9i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (310. - 537. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-651. - 376. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 913. iT - 7.89e6T^{2} \)
59 \( 1 + (1.14e3 - 659. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.06e3 + 3.57e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.34e3 - 5.78e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 2.88e3iT - 2.54e7T^{2} \)
73 \( 1 - 118.T + 2.83e7T^{2} \)
79 \( 1 + (3.91e3 - 6.78e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (4.40e3 + 2.54e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 1.02e4iT - 6.27e7T^{2} \)
97 \( 1 + (-440. + 762. i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−14.34117886649761885607402126904, −13.67200348907423560116070515627, −12.65281052721762021309178944172, −11.46458546168260317710342173513, −9.698814807846567241092731582914, −8.799231947355067661609636591350, −6.86989641249791471184993421394, −5.71834976080518090812464456928, −4.33505930058902202125304121455, −1.93384793846686714935110708219, 1.89577330874302287218566021595, 3.63302400226808293744246407589, 5.75022917604480505230925445539, 6.53301730978871213776011086250, 8.664709353843711208315338288080, 10.28900849031337738057134464273, 10.79643861409953863472504782049, 12.50030902019744859386286467216, 13.44872658136829483765828474296, 14.39953577570314830208570083691

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