Properties

Label 2-2736-19.18-c2-0-37
Degree $2$
Conductor $2736$
Sign $0.960 + 0.278i$
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  − 6.04·5-s + 0.880·7-s − 12.2·11-s − 7.16i·13-s − 15.7·17-s + (18.2 + 5.29i)19-s − 19.1·23-s + 11.5·25-s + 51.7i·29-s + 32.5i·31-s − 5.32·35-s − 25.3i·37-s − 19.0i·41-s − 39.5·43-s − 24.4·47-s + ⋯
L(s)  = 1  − 1.20·5-s + 0.125·7-s − 1.11·11-s − 0.551i·13-s − 0.926·17-s + (0.960 + 0.278i)19-s − 0.832·23-s + 0.460·25-s + 1.78i·29-s + 1.04i·31-s − 0.152·35-s − 0.686i·37-s − 0.465i·41-s − 0.920·43-s − 0.520·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.960 + 0.278i$
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: \((2,\ 2736,\ (\ :1),\ 0.960 + 0.278i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8232150857\)
\(L(\frac12)\) \(\approx\) \(0.8232150857\)
\(L(2)\) 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 \( 1 + (-18.2 - 5.29i)T \)
good5 \( 1 + 6.04T + 25T^{2} \)
7 \( 1 - 0.880T + 49T^{2} \)
11 \( 1 + 12.2T + 121T^{2} \)
13 \( 1 + 7.16iT - 169T^{2} \)
17 \( 1 + 15.7T + 289T^{2} \)
23 \( 1 + 19.1T + 529T^{2} \)
29 \( 1 - 51.7iT - 841T^{2} \)
31 \( 1 - 32.5iT - 961T^{2} \)
37 \( 1 + 25.3iT - 1.36e3T^{2} \)
41 \( 1 + 19.0iT - 1.68e3T^{2} \)
43 \( 1 + 39.5T + 1.84e3T^{2} \)
47 \( 1 + 24.4T + 2.20e3T^{2} \)
53 \( 1 - 13.1iT - 2.80e3T^{2} \)
59 \( 1 + 65.5iT - 3.48e3T^{2} \)
61 \( 1 - 22.2T + 3.72e3T^{2} \)
67 \( 1 + 18.7iT - 4.48e3T^{2} \)
71 \( 1 + 76.2iT - 5.04e3T^{2} \)
73 \( 1 - 85.4T + 5.32e3T^{2} \)
79 \( 1 + 150. iT - 6.24e3T^{2} \)
83 \( 1 - 29.0T + 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 - 127. iT - 9.40e3T^{2} \)
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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

−8.401431496590892089339440556241, −7.88315875731517725565245076132, −7.29549998695995797363316124637, −6.43718703298028229258528943259, −5.25634407724876777220517134923, −4.83622704839155137586564224899, −3.65519190109230164622471173548, −3.12327608142168806085864723126, −1.85211860420611905141739433604, −0.39391122653925277636771213659, 0.48284087719310112163941601555, 2.05316911484560094800107288456, 2.99666653887109392572837996108, 4.04622314309375110841756498946, 4.57731135678785763599735685803, 5.53190930383481799447921661546, 6.49390295346706315869121217749, 7.31526180450427395378424197284, 8.061471798189314520848913776344, 8.314087841362571685204032072569

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