Properties

Label 2-2736-19.18-c2-0-51
Degree $2$
Conductor $2736$
Sign $0.830 - 0.556i$
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  + 7.39·5-s − 3.28·7-s − 0.714·11-s + 25.4i·13-s + 26.0·17-s + (15.7 − 10.5i)19-s + 8.46·23-s + 29.6·25-s − 23.1i·29-s − 18.8i·31-s − 24.2·35-s − 48.5i·37-s + 35.3i·41-s + 24.5·43-s − 46.0·47-s + ⋯
L(s)  = 1  + 1.47·5-s − 0.469·7-s − 0.0649·11-s + 1.95i·13-s + 1.53·17-s + (0.830 − 0.556i)19-s + 0.367·23-s + 1.18·25-s − 0.797i·29-s − 0.608i·31-s − 0.693·35-s − 1.31i·37-s + 0.862i·41-s + 0.569·43-s − 0.980·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.830 - 0.556i$
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: \((2,\ 2736,\ (\ :1),\ 0.830 - 0.556i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.982140737\)
\(L(\frac12)\) \(\approx\) \(2.982140737\)
\(L(2)\) 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 \( 1 + (-15.7 + 10.5i)T \)
good5 \( 1 - 7.39T + 25T^{2} \)
7 \( 1 + 3.28T + 49T^{2} \)
11 \( 1 + 0.714T + 121T^{2} \)
13 \( 1 - 25.4iT - 169T^{2} \)
17 \( 1 - 26.0T + 289T^{2} \)
23 \( 1 - 8.46T + 529T^{2} \)
29 \( 1 + 23.1iT - 841T^{2} \)
31 \( 1 + 18.8iT - 961T^{2} \)
37 \( 1 + 48.5iT - 1.36e3T^{2} \)
41 \( 1 - 35.3iT - 1.68e3T^{2} \)
43 \( 1 - 24.5T + 1.84e3T^{2} \)
47 \( 1 + 46.0T + 2.20e3T^{2} \)
53 \( 1 - 51.2iT - 2.80e3T^{2} \)
59 \( 1 - 101. iT - 3.48e3T^{2} \)
61 \( 1 - 38.5T + 3.72e3T^{2} \)
67 \( 1 + 44.2iT - 4.48e3T^{2} \)
71 \( 1 + 74.3iT - 5.04e3T^{2} \)
73 \( 1 - 112.T + 5.32e3T^{2} \)
79 \( 1 - 125. iT - 6.24e3T^{2} \)
83 \( 1 + 95.5T + 6.88e3T^{2} \)
89 \( 1 - 28.0iT - 7.92e3T^{2} \)
97 \( 1 - 55.7iT - 9.40e3T^{2} \)
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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

−9.110357447144663897338031389337, −7.905206796843288769102832634569, −7.06146646606163297988950108935, −6.34611874413777567466489879198, −5.74500924073182389812048274139, −4.95232612865388466859993975118, −3.94281481325164339636705683254, −2.84647678530657696385511381420, −2.00035995485280845345911688336, −1.05209993051750320186607871022, 0.789224482242661836178583186361, 1.71463517325973325382029507665, 3.08547790025753216452016406608, 3.29367877260511241730055276797, 5.12873376786682025413110117494, 5.42528560443816648802857925129, 6.11176047386962873725684102987, 6.99079815588160348130902493076, 7.893932066579692974370657612194, 8.538881900258509256199293123288

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