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Properties

Label 1-3549-3549.185-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.986 - 0.163i$
Analytic cond. $16.4814$
Root an. cond. $16.4814$
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  + (0.632 − 0.774i)2-s + (−0.200 − 0.979i)4-s + (−0.919 − 0.391i)5-s + (−0.885 − 0.464i)8-s + (−0.885 + 0.464i)10-s + (0.354 − 0.935i)11-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s − 19-s + (−0.200 + 0.979i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.692 + 0.721i)25-s + (−0.987 − 0.160i)29-s + (−0.278 − 0.960i)31-s + (−0.278 + 0.960i)32-s + ⋯
L(s)  = 1  + (0.632 − 0.774i)2-s + (−0.200 − 0.979i)4-s + (−0.919 − 0.391i)5-s + (−0.885 − 0.464i)8-s + (−0.885 + 0.464i)10-s + (0.354 − 0.935i)11-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s − 19-s + (−0.200 + 0.979i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.692 + 0.721i)25-s + (−0.987 − 0.160i)29-s + (−0.278 − 0.960i)31-s + (−0.278 + 0.960i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004034769 - 0.08275951790i\)
\(L(\frac12)\) \(\approx\) \(1.004034769 - 0.08275951790i\)
\(L(1)\) \(\approx\) \(0.8832016405 - 0.5058838192i\)
\(L(1)\) \(\approx\) \(0.8832016405 - 0.5058838192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.632 - 0.774i)T \)
5 \( 1 + (-0.919 - 0.391i)T \)
11 \( 1 + (0.354 - 0.935i)T \)
17 \( 1 + (-0.845 + 0.534i)T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.987 - 0.160i)T \)
31 \( 1 + (-0.278 - 0.960i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (0.428 + 0.903i)T \)
43 \( 1 + (0.692 + 0.721i)T \)
47 \( 1 + (0.948 - 0.316i)T \)
53 \( 1 + (0.0402 + 0.999i)T \)
59 \( 1 + (-0.919 - 0.391i)T \)
61 \( 1 + (-0.885 + 0.464i)T \)
67 \( 1 + (-0.748 - 0.663i)T \)
71 \( 1 + (0.996 + 0.0804i)T \)
73 \( 1 + (-0.987 + 0.160i)T \)
79 \( 1 + (0.948 - 0.316i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.919 - 0.391i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.64942528595194303530125488647, −17.89001181478365601820059989924, −17.30353253618931575312028619037, −16.491769147324201875195363343891, −15.88206440787920697392090733067, −15.19268366172561336317404911248, −14.75622757546857843697011697461, −14.140558895946333573734106892978, −13.21915786752270969467358249017, −12.390753173570224287883326736435, −12.18800901538811650183740621004, −11.048505512288504138502847610518, −10.676647145925207059457266472320, −9.25079685435537366169459002861, −8.832760638672349241302998721307, −7.9229799591192348266981806861, −7.108903627498813732669371740, −6.90085227978921005487121541254, −5.955111707414004259439079873023, −4.95185671303132985320918518419, −4.296305758094913204290493616756, −3.81553845453280580254997861821, −2.78875229885229485597658311683, −2.05243232511212045421556270787, −0.28984512233548186142752635960, 0.84102676677761753524693402433, 1.709358335915898864175476705152, 2.744581474509099388018219259, 3.543593664627810345948226324358, 4.18721352963581463399988533781, 4.730323210476109558910884984318, 5.8202192782858278083090623062, 6.292064112606327127489240226267, 7.374830865192254485654183534719, 8.236766353506676304575877047355, 9.01312558692142884971525419207, 9.50475454328000482273423997438, 10.85116967022403888639738067494, 10.99100778377263808619356720977, 11.731383361556061601497122109013, 12.44252805830661808773900375295, 13.19550151858648606027075665746, 13.55662605958667584435188480041, 14.65967066425327681923666546893, 15.15943310998495022432211419338, 15.69252074118873997923084520698, 16.694762409860896733917844774769, 17.18717298320378836366504003257, 18.381234099767647663959791464088, 18.91517575395738829465678701866

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