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Properties

Label 2-2736-19.18-c2-0-38
Degree $2$
Conductor $2736$
Sign $0.159 - 0.987i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.79·5-s − 9.11·7-s + 18.2·11-s + 1.61i·13-s + 10.5·17-s + (−3.02 + 18.7i)19-s − 33.7·23-s + 35.7·25-s + 53.8i·29-s + 54.3i·31-s − 71.0·35-s − 20.2i·37-s − 13.2i·41-s − 45.1·43-s − 28.6·47-s + ⋯
L(s)  = 1  + 1.55·5-s − 1.30·7-s + 1.65·11-s + 0.124i·13-s + 0.622·17-s + (−0.159 + 0.987i)19-s − 1.46·23-s + 1.43·25-s + 1.85i·29-s + 1.75i·31-s − 2.03·35-s − 0.547i·37-s − 0.323i·41-s − 1.04·43-s − 0.609·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ 0.159 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.277668449\)
\(L(\frac12)\) \(\approx\) \(2.277668449\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (3.02 - 18.7i)T \)
good5 \( 1 - 7.79T + 25T^{2} \)
7 \( 1 + 9.11T + 49T^{2} \)
11 \( 1 - 18.2T + 121T^{2} \)
13 \( 1 - 1.61iT - 169T^{2} \)
17 \( 1 - 10.5T + 289T^{2} \)
23 \( 1 + 33.7T + 529T^{2} \)
29 \( 1 - 53.8iT - 841T^{2} \)
31 \( 1 - 54.3iT - 961T^{2} \)
37 \( 1 + 20.2iT - 1.36e3T^{2} \)
41 \( 1 + 13.2iT - 1.68e3T^{2} \)
43 \( 1 + 45.1T + 1.84e3T^{2} \)
47 \( 1 + 28.6T + 2.20e3T^{2} \)
53 \( 1 - 8.24iT - 2.80e3T^{2} \)
59 \( 1 + 73.0iT - 3.48e3T^{2} \)
61 \( 1 - 71.0T + 3.72e3T^{2} \)
67 \( 1 + 85.4iT - 4.48e3T^{2} \)
71 \( 1 - 113. iT - 5.04e3T^{2} \)
73 \( 1 + 68.1T + 5.32e3T^{2} \)
79 \( 1 - 99.4iT - 6.24e3T^{2} \)
83 \( 1 - 133.T + 6.88e3T^{2} \)
89 \( 1 - 89.2iT - 7.92e3T^{2} \)
97 \( 1 - 85.7iT - 9.40e3T^{2} \)
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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

−9.077566154816092085677620189450, −8.253546099118645951352159302496, −6.77778463132705762270162007754, −6.64828607672059950869226761473, −5.86152361096159459466158252740, −5.17160878207781514151232605529, −3.79410477887505663023314706414, −3.26712553788148684429794917591, −1.95462434398181619915639692402, −1.26941382374648975454093667034, 0.50776316856439185863841659991, 1.75074403536698719573045879252, 2.59631829853875994962570969877, 3.61852062756581471806610716774, 4.47187669619407981944892646942, 5.74839642269414715716539328360, 6.24325269126908154851563585042, 6.55230589089423319097716394541, 7.65553905811707544886574849922, 8.760168275547720346572749839122

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