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Properties

Label 1-3549-3549.17-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.968 + 0.248i$
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  + (0.120 − 0.992i)2-s + (−0.970 − 0.239i)4-s + (0.0402 − 0.999i)5-s + (−0.354 + 0.935i)8-s + (−0.987 − 0.160i)10-s + (−0.919 − 0.391i)11-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (−0.5 − 0.866i)19-s + (−0.278 + 0.960i)20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.996 − 0.0804i)25-s + (0.919 − 0.391i)29-s + (0.428 − 0.903i)31-s + (0.568 − 0.822i)32-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)2-s + (−0.970 − 0.239i)4-s + (0.0402 − 0.999i)5-s + (−0.354 + 0.935i)8-s + (−0.987 − 0.160i)10-s + (−0.919 − 0.391i)11-s + (0.885 + 0.464i)16-s + (−0.354 + 0.935i)17-s + (−0.5 − 0.866i)19-s + (−0.278 + 0.960i)20-s + (−0.5 + 0.866i)22-s − 23-s + (−0.996 − 0.0804i)25-s + (0.919 − 0.391i)29-s + (0.428 − 0.903i)31-s + (0.568 − 0.822i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3111519382 + 0.03920516470i\)
\(L(\frac12)\) \(\approx\) \(0.3111519382 + 0.03920516470i\)
\(L(1)\) \(\approx\) \(0.5933513435 - 0.4877838767i\)
\(L(1)\) \(\approx\) \(0.5933513435 - 0.4877838767i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.120 - 0.992i)T \)
5 \( 1 + (0.0402 - 0.999i)T \)
11 \( 1 + (-0.919 - 0.391i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.919 - 0.391i)T \)
31 \( 1 + (0.428 - 0.903i)T \)
37 \( 1 + (-0.568 - 0.822i)T \)
41 \( 1 + (-0.948 - 0.316i)T \)
43 \( 1 + (0.428 + 0.903i)T \)
47 \( 1 + (-0.278 + 0.960i)T \)
53 \( 1 + (-0.987 + 0.160i)T \)
59 \( 1 + (-0.885 + 0.464i)T \)
61 \( 1 + (0.632 - 0.774i)T \)
67 \( 1 + (-0.278 + 0.960i)T \)
71 \( 1 + (-0.200 - 0.979i)T \)
73 \( 1 + (0.799 - 0.600i)T \)
79 \( 1 + (0.278 - 0.960i)T \)
83 \( 1 + (0.748 + 0.663i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.845 + 0.534i)T \)
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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

−18.44457877890228506011426912193, −18.026473654719949240821339641865, −17.394961512058016856220165733512, −16.52295671742969754366962011845, −15.716723328292025138096570695437, −15.43676396932407619863976252743, −14.576729408858378778758013690092, −13.93052820567145530861811027498, −13.54875590105734758119664570956, −12.4800044577167099041223811600, −11.96153963788303253398772961243, −10.839070341300870563528248067, −10.10929824453595896969916921901, −9.72566687351005683920185987529, −8.46587983340829513304988007370, −8.10229773782967665986802898707, −7.10561351089362069633280117410, −6.77291882667342575370099974897, −5.92093654417280052016277837389, −5.15192311800553741849289815790, −4.427903624221354365909845969890, −3.46576538847041908620595552475, −2.784688585616405773791246494302, −1.74155612663351842077678111499, −0.10711634991378732672368176424, 0.82857280908149655064617461496, 1.864162663726140171983164074, 2.48526969257770380599733377642, 3.48752891879694410226798189453, 4.370630741786855515348934230145, 4.81153412163797821208558265331, 5.7255142020778133517276687977, 6.340690486676190715959200373713, 7.90374616344827783894840020998, 8.21418472672388483015556402405, 9.03209272935532974935695583707, 9.66554446490810552299767729286, 10.50770855693277069553177853102, 11.01225641165965238285004142642, 11.93279263043151445376706695815, 12.47779540459199250319935454713, 13.1974067245320583613641803491, 13.54856528786218616557796968803, 14.40069780623831487059135459101, 15.4033935588792953187711852245, 15.91120245749039154960547352282, 16.88399686658484589123239497633, 17.55995321964068449366528723436, 17.99139090226140351423769519963, 19.07035464270047442081744813898

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