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Properties

Label 2-54-9.4-c5-0-3
Degree $2$
Conductor $54$
Sign $0.687 + 0.726i$
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 + (39.2 − 67.9i)5-s + (−110. − 191. i)7-s − 63.9·8-s + 313.·10-s + (115. + 199. i)11-s + (385. − 668. i)13-s + (442. − 766. i)14-s + (−128 − 221. i)16-s + 769.·17-s − 383.·19-s + (627. + 1.08e3i)20-s + (−460. + 797. i)22-s + (193. − 334. i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.701 − 1.21i)5-s + (−0.852 − 1.47i)7-s − 0.353·8-s + 0.992·10-s + (0.286 + 0.496i)11-s + (0.633 − 1.09i)13-s + (0.603 − 1.04i)14-s + (−0.125 − 0.216i)16-s + 0.646·17-s − 0.243·19-s + (0.350 + 0.607i)20-s + (−0.202 + 0.351i)22-s + (0.0761 − 0.131i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.687 + 0.726i$
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: \((2,\ 54,\ (\ :5/2),\ 0.687 + 0.726i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.72576 - 0.743268i\)
\(L(\frac12)\) \(\approx\) \(1.72576 - 0.743268i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-39.2 + 67.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (110. + 191. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-115. - 199. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-385. + 668. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 769.T + 1.41e6T^{2} \)
19 \( 1 + 383.T + 2.47e6T^{2} \)
23 \( 1 + (-193. + 334. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (394. + 683. i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (1.60e3 - 2.78e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 2.46e3T + 6.93e7T^{2} \)
41 \( 1 + (4.62e3 - 8.00e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (5.31e3 + 9.20e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-976. - 1.69e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 3.25e4T + 4.18e8T^{2} \)
59 \( 1 + (1.19e4 - 2.06e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.88e4 + 3.25e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.15e4 + 1.99e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.60e4T + 1.80e9T^{2} \)
73 \( 1 - 6.51e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.54e4 - 6.13e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-2.76e4 - 4.78e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + 1.05e4T + 5.58e9T^{2} \)
97 \( 1 + (-4.14e4 - 7.17e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.94602770826498140050682143259, −13.18308988078549505385856967038, −12.48460846745286350242144571605, −10.44156059400442552615078487751, −9.393805993221192292701335499223, −7.982508334779297618930263347249, −6.58375197187125397640834436515, −5.21273974504600654710122549979, −3.75363302368189771128664216782, −0.876106706630546291042653284556, 2.18706982085957630808451691556, 3.39916734034771710567147270882, 5.75602412144763639672671663488, 6.56761237896317178985773244313, 8.912597345150982246980835433635, 9.858570001911458503479371411251, 11.13183782462818039328265360966, 12.11999758732983469725701322158, 13.40115493157863375799938900734, 14.36737395721558873043514192496

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