Properties

Label 2-2736-19.18-c2-0-16
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.725·5-s − 13.8·7-s − 20.3·11-s − 18.9·17-s + 19·19-s − 30·23-s − 24.4·25-s + 10.0·35-s − 53.8·43-s − 86.5·47-s + 142.·49-s + 14.7·55-s + 5.12·61-s − 112.·73-s + 281.·77-s + 90·83-s + 13.7·85-s − 13.7·95-s + 102·101-s + 21.7·115-s + 261.·119-s + ⋯
L(s)  = 1  − 0.145·5-s − 1.97·7-s − 1.85·11-s − 1.11·17-s + 19-s − 1.30·23-s − 0.978·25-s + 0.286·35-s − 1.25·43-s − 1.84·47-s + 2.90·49-s + 0.268·55-s + 0.0839·61-s − 1.53·73-s + 3.65·77-s + 1.08·83-s + 0.161·85-s − 0.145·95-s + 1.00·101-s + 0.189·115-s + 2.19·119-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $1$
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: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2240552783\)
\(L(\frac12)\) \(\approx\) \(0.2240552783\)
\(L(2)\) 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 \( 1 - 19T \)
good5 \( 1 + 0.725T + 25T^{2} \)
7 \( 1 + 13.8T + 49T^{2} \)
11 \( 1 + 20.3T + 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + 18.9T + 289T^{2} \)
23 \( 1 + 30T + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 53.8T + 1.84e3T^{2} \)
47 \( 1 + 86.5T + 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 5.12T + 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 112.T + 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 90T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−8.623196319287837903710861543735, −7.84766702032665860812898650828, −7.13188712623525053583645225339, −6.30792460053435373934037708723, −5.70521743749958004090292382849, −4.76468423156670947018962824839, −3.64629273637404745869130457410, −2.99909277442492232223557999549, −2.13122823598996888680509362040, −0.21634792284992863637742898794, 0.21634792284992863637742898794, 2.13122823598996888680509362040, 2.99909277442492232223557999549, 3.64629273637404745869130457410, 4.76468423156670947018962824839, 5.70521743749958004090292382849, 6.30792460053435373934037708723, 7.13188712623525053583645225339, 7.84766702032665860812898650828, 8.623196319287837903710861543735

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