Properties

Label 2-54-27.16-c1-0-0
Degree $2$
Conductor $54$
Sign $0.625 - 0.780i$
Analytic cond. $0.431192$
Root an. cond. $0.656652$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 + 0.642i)2-s + (−1.36 + 1.07i)3-s + (0.173 + 0.984i)4-s + (0.696 + 0.253i)5-s + (−1.73 − 0.0539i)6-s + (0.717 − 4.07i)7-s + (−0.500 + 0.866i)8-s + (0.703 − 2.91i)9-s + (0.370 + 0.641i)10-s + (−4.27 + 1.55i)11-s + (−1.29 − 1.15i)12-s + (0.662 − 0.556i)13-s + (3.16 − 2.65i)14-s + (−1.21 + 0.401i)15-s + (−0.939 + 0.342i)16-s + (2.17 + 3.77i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−0.785 + 0.618i)3-s + (0.0868 + 0.492i)4-s + (0.311 + 0.113i)5-s + (−0.706 − 0.0220i)6-s + (0.271 − 1.53i)7-s + (−0.176 + 0.306i)8-s + (0.234 − 0.972i)9-s + (0.117 + 0.202i)10-s + (−1.28 + 0.468i)11-s + (−0.372 − 0.333i)12-s + (0.183 − 0.154i)13-s + (0.846 − 0.709i)14-s + (−0.314 + 0.103i)15-s + (−0.234 + 0.0855i)16-s + (0.528 + 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :1/2),\ 0.625 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822422 + 0.395042i\)
\(L(\frac12)\) \(\approx\) \(0.822422 + 0.395042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (1.36 - 1.07i)T \)
good5 \( 1 + (-0.696 - 0.253i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.717 + 4.07i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (4.27 - 1.55i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.662 + 0.556i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.17 - 3.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.777 - 1.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.608 + 3.45i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.50 + 2.10i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.85 - 10.5i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.97 - 1.65i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.58 + 0.941i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.68 + 9.54i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 4.00T + 53T^{2} \)
59 \( 1 + (1.34 + 0.489i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.751 - 4.26i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-10.0 + 8.42i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.54 - 4.40i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.286 + 0.496i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.17 - 4.34i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-7.06 - 5.92i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-6.19 + 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.40 - 1.96i)T + (74.3 - 62.3i)T^{2} \)
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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

−15.60792211901933189114972986063, −14.49679005561854283864826627467, −13.38226561288601886899943686693, −12.27438631536301560065151817138, −10.65490836344145337633912221483, −10.20068395613446654696349418484, −8.002830441724452405031123696588, −6.65158643732470709345400657438, −5.22042601629672777934718492782, −3.94215172792586259298413553251, 2.39363991817582668294811212740, 5.24154657136006027338953748870, 5.88484242553135047079676747998, 7.80460859551780265837531810505, 9.481003735007022639138362787149, 11.06122960206069902803615442099, 11.83110003741093156904671035201, 12.88134730037421261247844071945, 13.71402322914293604695564151867, 15.26553877769395835152167961989

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