Properties

Label 2-2736-19.18-c2-0-57
Degree $2$
Conductor $2736$
Sign $0.315 + 0.948i$
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  − 4·5-s + 5·7-s − 10·11-s − 3.60i·13-s − 15·17-s + (6 + 18.0i)19-s + 35·23-s − 9·25-s + 18.0i·29-s − 36.0i·31-s − 20·35-s + 21.6i·37-s + 36.0i·41-s + 20·43-s + 10·47-s + ⋯
L(s)  = 1  − 0.800·5-s + 0.714·7-s − 0.909·11-s − 0.277i·13-s − 0.882·17-s + (0.315 + 0.948i)19-s + 1.52·23-s − 0.359·25-s + 0.621i·29-s − 1.16i·31-s − 0.571·35-s + 0.584i·37-s + 0.879i·41-s + 0.465·43-s + 0.212·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.315 + 0.948i$
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: \((2,\ 2736,\ (\ :1),\ 0.315 + 0.948i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.188989249\)
\(L(\frac12)\) \(\approx\) \(1.188989249\)
\(L(2)\) 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 \( 1 + (-6 - 18.0i)T \)
good5 \( 1 + 4T + 25T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 + 10T + 121T^{2} \)
13 \( 1 + 3.60iT - 169T^{2} \)
17 \( 1 + 15T + 289T^{2} \)
23 \( 1 - 35T + 529T^{2} \)
29 \( 1 - 18.0iT - 841T^{2} \)
31 \( 1 + 36.0iT - 961T^{2} \)
37 \( 1 - 21.6iT - 1.36e3T^{2} \)
41 \( 1 - 36.0iT - 1.68e3T^{2} \)
43 \( 1 - 20T + 1.84e3T^{2} \)
47 \( 1 - 10T + 2.20e3T^{2} \)
53 \( 1 + 75.7iT - 2.80e3T^{2} \)
59 \( 1 + 18.0iT - 3.48e3T^{2} \)
61 \( 1 + 40T + 3.72e3T^{2} \)
67 \( 1 - 39.6iT - 4.48e3T^{2} \)
71 \( 1 + 108. iT - 5.04e3T^{2} \)
73 \( 1 - 105T + 5.32e3T^{2} \)
79 \( 1 + 36.0iT - 6.24e3T^{2} \)
83 \( 1 + 40T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 122. iT - 9.40e3T^{2} \)
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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

−8.244707355527985290235880355161, −7.88290854629402990495908719076, −7.17180515432973766816353207980, −6.20282447320901445621730943715, −5.20876657982542643703634496794, −4.65524529142368771043941920414, −3.68903278694548071739308566010, −2.79311387547903411324198210592, −1.66178206422618766847379019366, −0.35367625412319439748779117245, 0.845452362997037263476895018683, 2.19910815342147758546937237887, 3.07504636853809730323061226255, 4.18366313702013218157772727958, 4.83080534861658485624373306665, 5.53071077033999279549734931785, 6.73744203742463694172657888398, 7.35145795643598342676412022753, 7.984555300306164340551570458060, 8.784036935276326818415262076723

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