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Properties

Label 2-3549-21.11-c0-0-5
Degree $2$
Conductor $3549$
Sign $0.832 + 0.553i$
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  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (−0.499 − 0.866i)16-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s + (1 − 1.73i)31-s + 0.999·36-s + (−0.5 − 0.866i)37-s − 43-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (−0.499 − 0.866i)16-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)28-s + (1 − 1.73i)31-s + 0.999·36-s + (−0.5 − 0.866i)37-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4243460122\)
\(L(\frac12)\) \(\approx\) \(0.4243460122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.917024041691691095407127315005, −8.032800074994184549775091142433, −7.13948889587554302539183732115, −6.40619324091849313121169154623, −5.60766262620111690979886834595, −4.72934255331099332147791562594, −4.00043491793406812645001250581, −3.38235173631294340438835810525, −2.50876337807693394982459645778, −0.30784408324835845337690575671, 1.07980475343351739084663867574, 2.11148989182615124736824773214, 3.24721283848882905357361516544, 4.36911468844692714771849069836, 5.22077223620258012334644498344, 5.94223156077839289999935208049, 6.53820953735487351487777747466, 7.01816395244996337594640737889, 8.298378793655290118622393516866, 8.566636341968201535802398108607

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