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Properties

Label 1-3549-3549.1886-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.380 - 0.924i$
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  + (0.200 + 0.979i)2-s + (−0.919 + 0.391i)4-s + (0.278 − 0.960i)5-s + (−0.568 − 0.822i)8-s + (0.996 + 0.0804i)10-s + (0.200 − 0.979i)11-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.5 − 0.866i)19-s + (0.120 + 0.992i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.845 − 0.534i)25-s + (0.748 − 0.663i)29-s + (0.0402 + 0.999i)31-s + (0.845 + 0.534i)32-s + ⋯
L(s)  = 1  + (0.200 + 0.979i)2-s + (−0.919 + 0.391i)4-s + (0.278 − 0.960i)5-s + (−0.568 − 0.822i)8-s + (0.996 + 0.0804i)10-s + (0.200 − 0.979i)11-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.5 − 0.866i)19-s + (0.120 + 0.992i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.845 − 0.534i)25-s + (0.748 − 0.663i)29-s + (0.0402 + 0.999i)31-s + (0.845 + 0.534i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.380 - 0.924i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1886, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.380 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266550352 - 0.8486657521i\)
\(L(\frac12)\) \(\approx\) \(1.266550352 - 0.8486657521i\)
\(L(1)\) \(\approx\) \(1.101062001 + 0.07325335418i\)
\(L(1)\) \(\approx\) \(1.101062001 + 0.07325335418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.200 + 0.979i)T \)
5 \( 1 + (0.278 - 0.960i)T \)
11 \( 1 + (0.200 - 0.979i)T \)
17 \( 1 + (0.428 - 0.903i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.748 - 0.663i)T \)
31 \( 1 + (0.0402 + 0.999i)T \)
37 \( 1 + (-0.845 + 0.534i)T \)
41 \( 1 + (-0.354 - 0.935i)T \)
43 \( 1 + (0.885 - 0.464i)T \)
47 \( 1 + (-0.919 - 0.391i)T \)
53 \( 1 + (-0.428 + 0.903i)T \)
59 \( 1 + (0.692 + 0.721i)T \)
61 \( 1 + (-0.428 - 0.903i)T \)
67 \( 1 + (0.799 - 0.600i)T \)
71 \( 1 + (0.354 + 0.935i)T \)
73 \( 1 + (0.200 - 0.979i)T \)
79 \( 1 + (-0.919 - 0.391i)T \)
83 \( 1 + (-0.354 + 0.935i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.970 + 0.239i)T \)
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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

−18.92821714627175221470119850765, −18.301007588142343400743860408366, −17.58271924315605472184027835775, −17.21544552959380002818344410887, −16.0022923383075258698671515619, −15.07529369268533516880617882400, −14.56064769323244512270876938932, −14.10753373249071558269453627295, −13.14857324141337085881124758190, −12.62944013673367632004375324378, −11.804750627129140528646455979403, −11.23195655133111259271051674362, −10.43414136531853877298675305688, −9.8815380856885356931932483679, −9.42795654741534364312937446529, −8.320471483222475416031251971834, −7.57919287382805640240893709720, −6.68213533141022664007991057177, −5.85873708264779606475245962904, −5.16599598016057977857929521484, −4.167626315627676948992181042937, −3.499416548553842336328956888305, −2.813735385739917461638328935996, −1.86465092704983010753605424391, −1.32926183603952571891738244960, 0.4597637175886177618139471978, 1.16171688897985652550427305552, 2.65456132367530414606520744402, 3.45051041782370005507622693248, 4.433987405295397703421904232516, 5.09187655689555573779722602658, 5.577474077862076526178335392692, 6.509333961581038404894667085066, 7.10427254574159967510691362836, 8.114474422015446302780447655942, 8.63245942890052054434282986524, 9.20777159417838901081395963545, 9.90539979198317704917843264260, 10.913403530343900761319545729550, 11.98162534879788857839167778414, 12.38260586015254656830287755562, 13.35240502790840347894021492220, 13.82126167086886911577595952078, 14.26737256679108667528877954534, 15.38501791214438319494307329404, 15.948699113785335912384431213721, 16.41708021587453665575959256337, 17.15596317600211313243282331834, 17.608532829504445716943654352346, 18.49049377109783694613413744102

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