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Properties

Label 2-3549-273.122-c0-0-3
Degree $2$
Conductor $3549$
Sign $0.763 - 0.645i$
Analytic cond. $1.77118$
Root an. cond. $1.33085$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)4-s − i·7-s + (0.499 + 0.866i)9-s + (0.499 + 0.866i)12-s + (0.499 + 0.866i)16-s + (−0.133 + 0.5i)19-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (0.5 − 0.866i)28-s + (1.36 − 0.366i)31-s + 0.999i·36-s + (−0.5 + 1.86i)37-s − 1.73i·43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)4-s − i·7-s + (0.499 + 0.866i)9-s + (0.499 + 0.866i)12-s + (0.499 + 0.866i)16-s + (−0.133 + 0.5i)19-s + (0.5 − 0.866i)21-s + (−0.866 − 0.5i)25-s + 0.999i·27-s + (0.5 − 0.866i)28-s + (1.36 − 0.366i)31-s + 0.999i·36-s + (−0.5 + 1.86i)37-s − 1.73i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.763 - 0.645i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1760, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.156947728\)
\(L(\frac12)\) \(\approx\) \(2.156947728\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
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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

−8.467482654682537256233776909777, −8.224515417755292801547829713750, −7.39800053113114581642090763855, −6.84243264972210066045982469751, −5.96960983158638657413887327335, −4.76939617025721734329560526040, −3.99492278977204482655334756746, −3.38069923798820802894288882083, −2.49807628446436336015765398928, −1.54132167699311601105641325056, 1.29292888654026365294802064565, 2.28374824693824292094815139105, 2.76652666260140134675378462976, 3.78752850608061866641817781884, 4.99699419022999316002492966693, 5.86601560726842506765778402557, 6.47348845057185527055172146407, 7.18980379388200915578803852445, 7.927650670350366899283112586007, 8.586625890859466504272731828197

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