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Properties

Label 1-3549-3549.68-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.610 - 0.791i$
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  + (0.748 + 0.663i)2-s + (0.120 + 0.992i)4-s + (0.278 + 0.960i)5-s + (−0.568 + 0.822i)8-s + (−0.428 + 0.903i)10-s + (−0.948 − 0.316i)11-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (0.5 − 0.866i)19-s + (−0.919 + 0.391i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.845 + 0.534i)25-s + (−0.948 + 0.316i)29-s + (0.0402 − 0.999i)31-s + (−0.885 − 0.464i)32-s + ⋯
L(s)  = 1  + (0.748 + 0.663i)2-s + (0.120 + 0.992i)4-s + (0.278 + 0.960i)5-s + (−0.568 + 0.822i)8-s + (−0.428 + 0.903i)10-s + (−0.948 − 0.316i)11-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (0.5 − 0.866i)19-s + (−0.919 + 0.391i)20-s + (−0.5 − 0.866i)22-s − 23-s + (−0.845 + 0.534i)25-s + (−0.948 + 0.316i)29-s + (0.0402 − 0.999i)31-s + (−0.885 − 0.464i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.610 - 0.791i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.610 - 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6669253952 - 0.3278139708i\)
\(L(\frac12)\) \(\approx\) \(0.6669253952 - 0.3278139708i\)
\(L(1)\) \(\approx\) \(1.094453491 + 0.5671097002i\)
\(L(1)\) \(\approx\) \(1.094453491 + 0.5671097002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.748 + 0.663i)T \)
5 \( 1 + (0.278 + 0.960i)T \)
11 \( 1 + (-0.948 - 0.316i)T \)
17 \( 1 + (0.568 - 0.822i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.948 + 0.316i)T \)
31 \( 1 + (0.0402 - 0.999i)T \)
37 \( 1 + (0.885 - 0.464i)T \)
41 \( 1 + (-0.632 - 0.774i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (-0.428 - 0.903i)T \)
59 \( 1 + (-0.970 - 0.239i)T \)
61 \( 1 + (0.996 - 0.0804i)T \)
67 \( 1 + (-0.919 + 0.391i)T \)
71 \( 1 + (-0.987 + 0.160i)T \)
73 \( 1 + (0.200 + 0.979i)T \)
79 \( 1 + (-0.919 + 0.391i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.692 - 0.721i)T \)
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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

−18.88918860247353200007043942899, −18.25261912286738489644287916502, −17.58796486394352485310203375066, −16.456334290709658633687248876111, −16.192367272830559521958187181029, −15.19220326928975062924242338539, −14.64467216859464553335218017963, −13.74813321510837488407270983716, −13.229435774394918201444211384934, −12.55842897380641480795661699183, −12.10808889682239098259328056526, −11.32046710765438783567688101411, −10.30454123204134904822053030403, −9.969605435636003154385883787527, −9.21677056663858418197912564678, −8.174944603664869270006043022360, −7.65819115344900323234705593321, −6.31013393442008486650328550357, −5.79311791489667199712483837855, −5.08099151187392692367496684882, −4.44124167991063915714919381565, −3.61815482444602385355066077377, −2.78216503470120756104621307736, −1.73945669789580283879144377684, −1.31208095352605790127114572317, 0.14672523791192883922757978778, 1.98442176097742122355546215193, 2.73409516508208045303781826520, 3.31652108043056130342418833578, 4.171254184666669458545428035887, 5.21706269913688804162174795276, 5.671607205956842855990873295164, 6.43665330102507873436183273491, 7.37086124962744552699771610279, 7.583838529006086375110942108305, 8.56741466734187956446955060357, 9.53466925804869345648864605398, 10.22043910733781209060877859369, 11.28583137644441214051721131603, 11.51576275415100691785559006165, 12.59611316722930484645548294428, 13.3359449933297960742724819982, 13.81583420267895090280075659181, 14.47739690396391700697564379907, 15.12725429481987465767726684509, 15.83246231216295889951547999126, 16.30676228720277322677983907897, 17.226699175865905550363128268363, 17.937579551882142421075621496400, 18.42398487286349154688611487402

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