Properties

Label 2-2736-19.18-c2-0-24
Degree $2$
Conductor $2736$
Sign $0.743 - 0.669i$
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  + 0.0118·5-s − 5.38·7-s − 20.1·11-s − 18.7i·13-s + 2.61·17-s + (−14.1 + 12.7i)19-s + 12.3·23-s − 24.9·25-s + 4.07i·29-s − 14.9i·31-s − 0.0637·35-s − 62.7i·37-s + 46.8i·41-s + 72.6·43-s − 81.3·47-s + ⋯
L(s)  = 1  + 0.00236·5-s − 0.768·7-s − 1.83·11-s − 1.44i·13-s + 0.153·17-s + (−0.743 + 0.669i)19-s + 0.535·23-s − 0.999·25-s + 0.140i·29-s − 0.482i·31-s − 0.00182·35-s − 1.69i·37-s + 1.14i·41-s + 1.68·43-s − 1.73·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.743 - 0.669i$
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: \((2,\ 2736,\ (\ :1),\ 0.743 - 0.669i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8541546797\)
\(L(\frac12)\) \(\approx\) \(0.8541546797\)
\(L(2)\) 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 \( 1 + (14.1 - 12.7i)T \)
good5 \( 1 - 0.0118T + 25T^{2} \)
7 \( 1 + 5.38T + 49T^{2} \)
11 \( 1 + 20.1T + 121T^{2} \)
13 \( 1 + 18.7iT - 169T^{2} \)
17 \( 1 - 2.61T + 289T^{2} \)
23 \( 1 - 12.3T + 529T^{2} \)
29 \( 1 - 4.07iT - 841T^{2} \)
31 \( 1 + 14.9iT - 961T^{2} \)
37 \( 1 + 62.7iT - 1.36e3T^{2} \)
41 \( 1 - 46.8iT - 1.68e3T^{2} \)
43 \( 1 - 72.6T + 1.84e3T^{2} \)
47 \( 1 + 81.3T + 2.20e3T^{2} \)
53 \( 1 - 32.5iT - 2.80e3T^{2} \)
59 \( 1 - 48.9iT - 3.48e3T^{2} \)
61 \( 1 - 81.6T + 3.72e3T^{2} \)
67 \( 1 - 64.8iT - 4.48e3T^{2} \)
71 \( 1 - 18.7iT - 5.04e3T^{2} \)
73 \( 1 - 117.T + 5.32e3T^{2} \)
79 \( 1 + 80.1iT - 6.24e3T^{2} \)
83 \( 1 - 61.0T + 6.88e3T^{2} \)
89 \( 1 + 132. iT - 7.92e3T^{2} \)
97 \( 1 + 91.8iT - 9.40e3T^{2} \)
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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

−8.631802899463379919411966464651, −7.78950235866084030780999048551, −7.55098513657852047918451548849, −6.25363604071413741210953026261, −5.69377830776921860328144515129, −4.99384442904117786727706469912, −3.83033133935496273711642433346, −2.97570756773410130673234974983, −2.25495479228044474962071313124, −0.60903662781626928856494798744, 0.31268727389703265021368548292, 1.95448912811439294130537566844, 2.74662378065112818661611539577, 3.71004189939303770500484416100, 4.72377442470885268603068198548, 5.37043446584702001846009212321, 6.45630297626511251470968207863, 6.88345735573049593887236628710, 7.890603683689396446523386254041, 8.474720469722563287279479528983

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