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Properties

Label 2-54-9.4-c3-0-1
Degree $2$
Conductor $54$
Sign $-0.173 - 0.984i$
Analytic cond. $3.18610$
Root an. cond. $1.78496$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−4.5 + 7.79i)5-s + (15.5 + 26.8i)7-s − 7.99·8-s − 18·10-s + (−7.5 − 12.9i)11-s + (18.5 − 32.0i)13-s + (−31 + 53.6i)14-s + (−8 − 13.8i)16-s + 42·17-s − 28·19-s + (−18 − 31.1i)20-s + (15 − 25.9i)22-s + (97.5 − 168. i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.402 + 0.697i)5-s + (0.836 + 1.44i)7-s − 0.353·8-s − 0.569·10-s + (−0.205 − 0.356i)11-s + (0.394 − 0.683i)13-s + (−0.591 + 1.02i)14-s + (−0.125 − 0.216i)16-s + 0.599·17-s − 0.338·19-s + (−0.201 − 0.348i)20-s + (0.145 − 0.251i)22-s + (0.883 − 1.53i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(3.18610\)
Root analytic conductor: \(1.78496\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00245 + 1.19468i\)
\(L(\frac12)\) \(\approx\) \(1.00245 + 1.19468i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (4.5 - 7.79i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-15.5 - 26.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (7.5 + 12.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-18.5 + 32.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 42T + 4.91e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 + (-97.5 + 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-55.5 - 96.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-102.5 + 177. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 166T + 5.06e4T^{2} \)
41 \( 1 + (130.5 - 226. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-88.5 - 153. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 114T + 1.48e5T^{2} \)
59 \( 1 + (-79.5 + 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (95.5 + 165. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-210.5 + 364. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 156T + 3.57e5T^{2} \)
73 \( 1 - 182T + 3.89e5T^{2} \)
79 \( 1 + (566.5 + 981. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (541.5 + 937. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (-450.5 - 780. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.02788543344782576002839256292, −14.49245214391693565550972853206, −12.93094188716750862086067381351, −11.83405778830498556614481465217, −10.72487723710863937749488304131, −8.803566294988039478729661914623, −7.894905011763843281176777610457, −6.29245567947477732017370394080, −5.02723470767631444413009937051, −2.94885710203272923111949208319, 1.23133572582289642226086381519, 3.91088914816389999323346118898, 5.00133114795564324482843121360, 7.18133267313911356192398236147, 8.534537828215533188585232935729, 10.09586647798347804500733231754, 11.16313383320124276922929689355, 12.19129070562825900658008414996, 13.46427293191319197544746231442, 14.20298220101703776844239061329

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