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Properties

Label 1-1287-1287.479-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.803 + 0.595i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.406 − 0.913i)5-s + (−0.743 + 0.669i)7-s + (0.951 − 0.309i)8-s + (−0.5 + 0.866i)10-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (0.994 − 0.104i)20-s + (−0.5 − 0.866i)23-s + (−0.669 + 0.743i)25-s + (−0.406 − 0.913i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.406 − 0.913i)5-s + (−0.743 + 0.669i)7-s + (0.951 − 0.309i)8-s + (−0.5 + 0.866i)10-s + (0.978 + 0.207i)14-s + (−0.809 − 0.587i)16-s + (0.104 − 0.994i)17-s + (−0.743 − 0.669i)19-s + (0.994 − 0.104i)20-s + (−0.5 − 0.866i)23-s + (−0.669 + 0.743i)25-s + (−0.406 − 0.913i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.803 + 0.595i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.803 + 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1782136082 - 0.5400075506i\)
\(L(\frac12)\) \(\approx\) \(-0.1782136082 - 0.5400075506i\)
\(L(1)\) \(\approx\) \(0.5234256754 - 0.3360129721i\)
\(L(1)\) \(\approx\) \(0.5234256754 - 0.3360129721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (-0.406 - 0.913i)T \)
7 \( 1 + (-0.743 + 0.669i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.994 - 0.104i)T \)
37 \( 1 + (0.743 - 0.669i)T \)
41 \( 1 + (0.743 + 0.669i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.743 + 0.669i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.994 + 0.104i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.994 - 0.104i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.994 - 0.104i)T \)
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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

−21.4225500441240438865472134033, −20.04628694698145448338031410137, −19.620183860328218533538128068685, −18.88441356520816263531254636130, −18.31289099213943054703856544392, −17.26705098755224930558552926122, −16.8428709723088008482223931145, −15.808813402721911081719995105954, −15.35849621304808147631742362533, −14.45085601050051741079739820194, −13.87743584222789511029992275513, −12.95113593498129329244516278051, −11.88063007340188664692287806593, −10.700608659704118542871597803573, −10.37347729910796388943377156218, −9.59872614064821213870951661596, −8.46374123719426504969554094614, −7.796950038908736543310674954158, −6.93638174123085740086214304582, −6.403159127114306938315168907525, −5.58299562793643463818700603354, −4.20428813159217695109761168151, −3.588018022708497615313703217061, −2.265615304007654913751559592888, −1.01006575210242429117244981037, 0.20169873145854220155842717508, 0.86201734306987290023725359311, 2.2764361378443761058457119059, 2.88510281344596173448677830086, 4.12212476687885365734634611723, 4.70242342688434425223750151799, 5.927014890912469771899755870071, 6.98372646894831081903902373737, 8.0555470771292274317396491341, 8.62290671334820121913016523181, 9.45030252494420499881303179419, 9.93815791263009147334651631670, 11.1654745115937038434297158014, 11.7950485580891575761948995984, 12.53712187114068411360161122606, 13.04251270368778180943447007899, 13.90657103237524953400577325558, 15.2406103473235383886434539895, 16.0374901710157839337584969727, 16.529542451108979632794595833194, 17.39232220138903472049002031849, 18.233087248330869942341973549071, 18.968299505007972224538314861513, 19.67733302799906674351912377070, 20.13984926431944511747843588235

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