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Properties

Label 2-3549-3549.2447-c0-0-0
Degree $2$
Conductor $3549$
Sign $-0.900 - 0.435i$
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  + (−0.970 + 0.239i)3-s + (−0.996 − 0.0804i)4-s + (−0.919 − 0.391i)7-s + (0.885 − 0.464i)9-s + (0.987 − 0.160i)12-s + (−0.5 − 0.866i)13-s + (0.987 + 0.160i)16-s − 0.0805·19-s + (0.987 + 0.160i)21-s + (0.948 − 0.316i)25-s + (−0.748 + 0.663i)27-s + (0.885 + 0.464i)28-s + (−0.227 − 1.11i)31-s + (−0.919 + 0.391i)36-s + (0.368 + 1.80i)37-s + ⋯
L(s)  = 1  + (−0.970 + 0.239i)3-s + (−0.996 − 0.0804i)4-s + (−0.919 − 0.391i)7-s + (0.885 − 0.464i)9-s + (0.987 − 0.160i)12-s + (−0.5 − 0.866i)13-s + (0.987 + 0.160i)16-s − 0.0805·19-s + (0.987 + 0.160i)21-s + (0.948 − 0.316i)25-s + (−0.748 + 0.663i)27-s + (0.885 + 0.464i)28-s + (−0.227 − 1.11i)31-s + (−0.919 + 0.391i)36-s + (0.368 + 1.80i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.900 - 0.435i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03754074155\)
\(L(\frac12)\) \(\approx\) \(0.03754074155\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.970 - 0.239i)T \)
7 \( 1 + (0.919 + 0.391i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.996 + 0.0804i)T^{2} \)
5 \( 1 + (-0.948 + 0.316i)T^{2} \)
11 \( 1 + (-0.568 - 0.822i)T^{2} \)
17 \( 1 + (-0.692 + 0.721i)T^{2} \)
19 \( 1 + 0.0805T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.428 - 0.903i)T^{2} \)
31 \( 1 + (0.227 + 1.11i)T + (-0.919 + 0.391i)T^{2} \)
37 \( 1 + (-0.368 - 1.80i)T + (-0.919 + 0.391i)T^{2} \)
41 \( 1 + (0.0402 - 0.999i)T^{2} \)
43 \( 1 + (1.74 - 0.582i)T + (0.799 - 0.600i)T^{2} \)
47 \( 1 + (0.632 + 0.774i)T^{2} \)
53 \( 1 + (-0.278 - 0.960i)T^{2} \)
59 \( 1 + (-0.948 + 0.316i)T^{2} \)
61 \( 1 + (0.240 + 1.97i)T + (-0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.850 + 1.23i)T + (-0.354 + 0.935i)T^{2} \)
71 \( 1 + (0.845 + 0.534i)T^{2} \)
73 \( 1 + (1.66 - 1.05i)T + (0.428 - 0.903i)T^{2} \)
79 \( 1 + (0.832 - 1.75i)T + (-0.632 - 0.774i)T^{2} \)
83 \( 1 + (-0.885 - 0.464i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.987 + 0.160i)T + (0.948 + 0.316i)T^{2} \)
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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

−9.274487088531596785670758821103, −8.334678331231456602800242733836, −7.57403258559581815678976340863, −6.62232866070877044556038824792, −6.08138027004855291878940299719, −5.15342411381991608545564626723, −4.66111614344767499891021041057, −3.76341292208534011568088325860, −2.95893985789382767825603261463, −1.13219298096514167450911437235, 0.03065058524348910976588908448, 1.53845474935160480028310769381, 2.90473293753458976128855288043, 3.95453477220050349239184277659, 4.68798731776684374735136332193, 5.43195459173334071871400558686, 6.07115434860505892209375824874, 6.98776728012881218484080208388, 7.42247195415284075645782410360, 8.767727120710493022838931258825

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