Properties

Label 2-2736-19.18-c2-0-65
Degree $2$
Conductor $2736$
Sign $-0.431 + 0.902i$
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  − 7.51·5-s + 7.72·7-s + 9.01·11-s + 10.6i·13-s − 31.2·17-s + (8.20 − 17.1i)19-s − 4.85·23-s + 31.4·25-s − 23.2i·29-s + 7.74i·31-s − 58.0·35-s + 21.4i·37-s + 32.7i·41-s − 7.82·43-s + 50.7·47-s + ⋯
L(s)  = 1  − 1.50·5-s + 1.10·7-s + 0.819·11-s + 0.818i·13-s − 1.83·17-s + (0.431 − 0.902i)19-s − 0.211·23-s + 1.25·25-s − 0.800i·29-s + 0.249i·31-s − 1.65·35-s + 0.579i·37-s + 0.798i·41-s − 0.182·43-s + 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.431 + 0.902i$
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: \((2,\ 2736,\ (\ :1),\ -0.431 + 0.902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7491969583\)
\(L(\frac12)\) \(\approx\) \(0.7491969583\)
\(L(2)\) 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 \( 1 + (-8.20 + 17.1i)T \)
good5 \( 1 + 7.51T + 25T^{2} \)
7 \( 1 - 7.72T + 49T^{2} \)
11 \( 1 - 9.01T + 121T^{2} \)
13 \( 1 - 10.6iT - 169T^{2} \)
17 \( 1 + 31.2T + 289T^{2} \)
23 \( 1 + 4.85T + 529T^{2} \)
29 \( 1 + 23.2iT - 841T^{2} \)
31 \( 1 - 7.74iT - 961T^{2} \)
37 \( 1 - 21.4iT - 1.36e3T^{2} \)
41 \( 1 - 32.7iT - 1.68e3T^{2} \)
43 \( 1 + 7.82T + 1.84e3T^{2} \)
47 \( 1 - 50.7T + 2.20e3T^{2} \)
53 \( 1 + 66.8iT - 2.80e3T^{2} \)
59 \( 1 - 86.1iT - 3.48e3T^{2} \)
61 \( 1 + 3.46T + 3.72e3T^{2} \)
67 \( 1 - 14.0iT - 4.48e3T^{2} \)
71 \( 1 + 35.9iT - 5.04e3T^{2} \)
73 \( 1 - 79.1T + 5.32e3T^{2} \)
79 \( 1 + 61.9iT - 6.24e3T^{2} \)
83 \( 1 + 143.T + 6.88e3T^{2} \)
89 \( 1 - 53.0iT - 7.92e3T^{2} \)
97 \( 1 + 124. iT - 9.40e3T^{2} \)
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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

−8.550110634830951586212480877511, −7.62053716947542980991129724418, −7.01621781252144155880753481690, −6.32453495619778065687686605850, −4.92634714605359832133155676939, −4.39409856405281768890099157280, −3.87632130500471085606617969180, −2.59591698155896926746913014538, −1.47767277041475737609745437005, −0.20530407607212001102468875543, 1.01132575221340791022013073678, 2.19233036291522150580167728350, 3.49567701325108314973182585090, 4.12939009425740469979860321721, 4.79219612369201928032549436770, 5.74977119471524573370124296868, 6.83854491331714627826929744878, 7.47105767177418116476585655357, 8.173400010145739487453331488688, 8.632351000599981157250073786012

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