Properties

Label 2-2736-19.18-c2-0-19
Degree $2$
Conductor $2736$
Sign $-0.800 - 0.598i$
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  − 6.80·5-s + 9.74·7-s − 20.3·11-s + 24.2i·13-s + 21.2·17-s + (15.2 + 11.3i)19-s + 20.7·23-s + 21.3·25-s + 21.9i·29-s − 40.0i·31-s − 66.3·35-s − 44.5i·37-s + 17.7i·41-s − 4.69·43-s − 1.15·47-s + ⋯
L(s)  = 1  − 1.36·5-s + 1.39·7-s − 1.85·11-s + 1.86i·13-s + 1.24·17-s + (0.800 + 0.598i)19-s + 0.900·23-s + 0.853·25-s + 0.755i·29-s − 1.29i·31-s − 1.89·35-s − 1.20i·37-s + 0.432i·41-s − 0.109·43-s − 0.0244·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.800 - 0.598i$
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: \((2,\ 2736,\ (\ :1),\ -0.800 - 0.598i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8733915362\)
\(L(\frac12)\) \(\approx\) \(0.8733915362\)
\(L(2)\) 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 \( 1 + (-15.2 - 11.3i)T \)
good5 \( 1 + 6.80T + 25T^{2} \)
7 \( 1 - 9.74T + 49T^{2} \)
11 \( 1 + 20.3T + 121T^{2} \)
13 \( 1 - 24.2iT - 169T^{2} \)
17 \( 1 - 21.2T + 289T^{2} \)
23 \( 1 - 20.7T + 529T^{2} \)
29 \( 1 - 21.9iT - 841T^{2} \)
31 \( 1 + 40.0iT - 961T^{2} \)
37 \( 1 + 44.5iT - 1.36e3T^{2} \)
41 \( 1 - 17.7iT - 1.68e3T^{2} \)
43 \( 1 + 4.69T + 1.84e3T^{2} \)
47 \( 1 + 1.15T + 2.20e3T^{2} \)
53 \( 1 + 75.9iT - 2.80e3T^{2} \)
59 \( 1 - 0.973iT - 3.48e3T^{2} \)
61 \( 1 + 77.0T + 3.72e3T^{2} \)
67 \( 1 - 33.2iT - 4.48e3T^{2} \)
71 \( 1 - 91.7iT - 5.04e3T^{2} \)
73 \( 1 + 97.9T + 5.32e3T^{2} \)
79 \( 1 - 43.0iT - 6.24e3T^{2} \)
83 \( 1 - 55.7T + 6.88e3T^{2} \)
89 \( 1 - 74.9iT - 7.92e3T^{2} \)
97 \( 1 - 55.2iT - 9.40e3T^{2} \)
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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

−8.728243065748875200835614384397, −7.928690429201775516754432335002, −7.68529193358491286685948583505, −7.02180034078121593170930310522, −5.60649708477216985371817037937, −4.99268643585350883071211738520, −4.27963722821050097893539570442, −3.43157662421831674385608944453, −2.29473742710039702617050824702, −1.17991540694079081491584833499, 0.23700980254556763050100147727, 1.21268494102661488022362341427, 2.90800009764284977605104246317, 3.21695225664950769407270149132, 4.67391513931177492211086084627, 5.07086968865218500537467169358, 5.73349137383493712830942389234, 7.36300391965827230752028934708, 7.73912536952151852118219727408, 8.024542610229648072500918732974

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