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Properties

Label 1-3549-3549.776-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.0340 - 0.999i$
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  + (−0.692 − 0.721i)2-s + (−0.0402 + 0.999i)4-s + (0.568 + 0.822i)5-s + (0.748 − 0.663i)8-s + (0.200 − 0.979i)10-s + (−0.278 − 0.960i)11-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (0.5 − 0.866i)19-s + (−0.845 + 0.534i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.354 + 0.935i)25-s + (−0.692 − 0.721i)29-s + (0.354 + 0.935i)31-s + (0.632 + 0.774i)32-s + ⋯
L(s)  = 1  + (−0.692 − 0.721i)2-s + (−0.0402 + 0.999i)4-s + (0.568 + 0.822i)5-s + (0.748 − 0.663i)8-s + (0.200 − 0.979i)10-s + (−0.278 − 0.960i)11-s + (−0.996 − 0.0804i)16-s + (−0.200 − 0.979i)17-s + (0.5 − 0.866i)19-s + (−0.845 + 0.534i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.354 + 0.935i)25-s + (−0.692 − 0.721i)29-s + (0.354 + 0.935i)31-s + (0.632 + 0.774i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.0340 - 0.999i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (776, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.0340 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7935610205 - 0.7669667461i\)
\(L(\frac12)\) \(\approx\) \(0.7935610205 - 0.7669667461i\)
\(L(1)\) \(\approx\) \(0.7826541273 - 0.2344540347i\)
\(L(1)\) \(\approx\) \(0.7826541273 - 0.2344540347i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.692 - 0.721i)T \)
5 \( 1 + (0.568 + 0.822i)T \)
11 \( 1 + (-0.278 - 0.960i)T \)
17 \( 1 + (-0.200 - 0.979i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.692 - 0.721i)T \)
31 \( 1 + (0.354 + 0.935i)T \)
37 \( 1 + (-0.632 + 0.774i)T \)
41 \( 1 + (0.799 - 0.600i)T \)
43 \( 1 + (-0.632 - 0.774i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
53 \( 1 + (0.748 - 0.663i)T \)
59 \( 1 + (-0.996 + 0.0804i)T \)
61 \( 1 + (-0.948 + 0.316i)T \)
67 \( 1 + (-0.0402 - 0.999i)T \)
71 \( 1 + (0.919 + 0.391i)T \)
73 \( 1 + (0.970 + 0.239i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.428 - 0.903i)T \)
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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

−18.61078657644836299834494249477, −18.12864342118754619056559167287, −17.43405100712844144290321834382, −16.7699300705454352758170196769, −16.43915795191313166519284484607, −15.461933346727865582973616584360, −14.93377066717464442082902037106, −14.21820211528283950328032845022, −13.42465028602460347525531789291, −12.69324901265635023332515520800, −12.10776516954967808852456779705, −10.867169720969554450492736407404, −10.37689351075242192795378864373, −9.52927872899573706171514827917, −9.15302182890039387735679737846, −8.22515651811160349434360548810, −7.74374479865979010085630959483, −6.81966605933924086973846987450, −6.06954092307416887084818491252, −5.41697568330842492058001388967, −4.71100989300823336788006867206, −3.94559922311135292271816773808, −2.420246691395159001320258387856, −1.73731628530046727958598256110, −0.94479573835851705781501907159, 0.47391408798749469318398293785, 1.51140896866975887354030378699, 2.476032640838647686080798865506, 3.04147022992581770211444457290, 3.65786256372188673857550437919, 4.8788928545798096581332421802, 5.6506644615126610730048226381, 6.70278071437323548089316867844, 7.24742482441523315440142944634, 7.99316130436395066479508346299, 9.025924473525071892874662085142, 9.36174200401681619005784779370, 10.24755253531760941445837290290, 10.87239014250400445522163868075, 11.40123898377400520336789436379, 12.02957984759831815290200744492, 13.1129187653969368444932002790, 13.72330539749735777295860986055, 14.01627680919392019838209198023, 15.42438488766747180913093376205, 15.72446842915767296804147203263, 16.82689332398516408400737723537, 17.26928232564519972634262043716, 18.1227463805626136168550498605, 18.45849947950044024117095861980

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