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Properties

Label 1-3549-3549.152-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.497 - 0.867i$
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  + (−0.799 − 0.600i)2-s + (0.278 + 0.960i)4-s + (−0.845 − 0.534i)5-s + (0.354 − 0.935i)8-s + (0.354 + 0.935i)10-s + (−0.120 − 0.992i)11-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s − 19-s + (0.278 − 0.960i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (0.428 + 0.903i)25-s + (0.919 − 0.391i)29-s + (0.996 − 0.0804i)31-s + (0.996 + 0.0804i)32-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)2-s + (0.278 + 0.960i)4-s + (−0.845 − 0.534i)5-s + (0.354 − 0.935i)8-s + (0.354 + 0.935i)10-s + (−0.120 − 0.992i)11-s + (−0.845 + 0.534i)16-s + (0.987 − 0.160i)17-s − 19-s + (0.278 − 0.960i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (0.428 + 0.903i)25-s + (0.919 − 0.391i)29-s + (0.996 − 0.0804i)31-s + (0.996 + 0.0804i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.497 - 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7999424503 - 0.4633530141i\)
\(L(\frac12)\) \(\approx\) \(0.7999424503 - 0.4633530141i\)
\(L(1)\) \(\approx\) \(0.6310610408 - 0.2437063122i\)
\(L(1)\) \(\approx\) \(0.6310610408 - 0.2437063122i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.799 - 0.600i)T \)
5 \( 1 + (-0.845 - 0.534i)T \)
11 \( 1 + (-0.120 - 0.992i)T \)
17 \( 1 + (0.987 - 0.160i)T \)
19 \( 1 - T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.919 - 0.391i)T \)
31 \( 1 + (0.996 - 0.0804i)T \)
37 \( 1 + (-0.996 + 0.0804i)T \)
41 \( 1 + (0.948 + 0.316i)T \)
43 \( 1 + (0.428 + 0.903i)T \)
47 \( 1 + (0.692 + 0.721i)T \)
53 \( 1 + (0.632 + 0.774i)T \)
59 \( 1 + (-0.845 - 0.534i)T \)
61 \( 1 + (0.354 + 0.935i)T \)
67 \( 1 + (-0.970 + 0.239i)T \)
71 \( 1 + (0.200 + 0.979i)T \)
73 \( 1 + (0.919 + 0.391i)T \)
79 \( 1 + (0.692 + 0.721i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
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

−18.804451480129591756955237884391, −18.16173557258805036876106846705, −17.38708883024646385014534313435, −16.94174056031726438843022816164, −15.9317448581252445297048371570, −15.509074082592973978742033988944, −14.89705292390934528972006736674, −14.33814642174236952221817789546, −13.50086365347678144833074564098, −12.25036029259648620824165145257, −11.991695654832188653153290508901, −10.80251894157322822446192266192, −10.50416839853523838508750522244, −9.7242268953584830903274264985, −8.86162056859506524124899959275, −8.158684413153355418979006358606, −7.484963940087957983168845304313, −6.96871878792066731429641957347, −6.2543970462572128164899194041, −5.28085612801429803436995015023, −4.545512938507633068951845494452, −3.61256027446018346079446749926, −2.60573086068625307932857557239, −1.73599294173912099990775191336, −0.62898524172307582762557955306, 0.67824074230380220454283769736, 1.1979187703467554787546082971, 2.57893917670088984234206921322, 3.10210203472413944901812597855, 4.076647331217908780123989434741, 4.61845352833569600320072227916, 5.80651825156460867102882803324, 6.67653337043333209892456165811, 7.5645449886199587723383575039, 8.25493019747725435097779105259, 8.6146124200752605101041780440, 9.40905319189701366563525866009, 10.3438852210194796207566351438, 10.89400583465851788739532216664, 11.591325703720969126345316058188, 12.297799849087635044463385187956, 12.7286665837309797054727036383, 13.61975833551218130810626044588, 14.45864439547457396792045030041, 15.511262724065087550226414782719, 15.97187709180858318977571719414, 16.77234116419522190358518207122, 17.05525973688485304386967507480, 18.05340431086036689794600508523, 18.88440479743049183045882508063

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