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Properties

Label 2-54-9.4-c1-0-0
Degree $2$
Conductor $54$
Sign $0.766 - 0.642i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)7-s − 0.999·8-s + (−1.5 − 2.59i)11-s + (−1 + 1.73i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s + 3·17-s − 19-s + (1.5 − 2.59i)22-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s − 1.99·26-s + 1.99·28-s + (3 + 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.377 − 0.654i)7-s − 0.353·8-s + (−0.452 − 0.783i)11-s + (−0.277 + 0.480i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s + 0.727·17-s − 0.229·19-s + (0.319 − 0.553i)22-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s − 0.392·26-s + 0.377·28-s + (0.557 + 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54\)    =    \(2 \cdot 3^{3}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{54} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 54,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877258 + 0.319296i\)
\(L(\frac12)\) \(\approx\) \(0.877258 + 0.319296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.59861712213794375411483361128, −14.27482924604470976704252044001, −13.51017109175237931670563352488, −12.33645744571137792579080469536, −10.91361990419339911954793543618, −9.517129125896991291576106911816, −8.035203487704401873263663516929, −6.82058941932479782202029435549, −5.35381362714877193537420366400, −3.57176631322221135410380789907, 2.69831856592917915664558595168, 4.71102187921828010545904359307, 6.24346619468319563425589462519, 8.109217585919713043018362675136, 9.644156657279471627612142445110, 10.57427835288387395140515205717, 12.17511820172585237543301203042, 12.65616926729159557398117926597, 14.09692654269574911820876482947, 15.12688103669667857937205546939

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