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Properties

Label 1-1287-1287.1168-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.629 + 0.776i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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 + (0.994 − 0.104i)5-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + (−0.5 + 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.913 + 0.406i)17-s + (0.207 + 0.978i)19-s + (−0.406 − 0.913i)20-s + (0.5 + 0.866i)23-s + (0.978 − 0.207i)25-s + (−0.994 + 0.104i)28-s + (−0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.994 − 0.104i)5-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s + (−0.5 + 0.866i)10-s + (0.669 + 0.743i)14-s + (−0.809 + 0.587i)16-s + (0.913 + 0.406i)17-s + (0.207 + 0.978i)19-s + (−0.406 − 0.913i)20-s + (0.5 + 0.866i)23-s + (0.978 − 0.207i)25-s + (−0.994 + 0.104i)28-s + (−0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.629 + 0.776i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.629 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.316102689 + 0.6270823147i\)
\(L(\frac12)\) \(\approx\) \(1.316102689 + 0.6270823147i\)
\(L(1)\) \(\approx\) \(0.9794339213 + 0.2843034899i\)
\(L(1)\) \(\approx\) \(0.9794339213 + 0.2843034899i\)

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 \( 1 + (0.994 - 0.104i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.207 + 0.978i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (0.406 + 0.913i)T \)
37 \( 1 + (-0.207 + 0.978i)T \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.406 + 0.913i)T \)
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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

−20.84978629539309458490122349234, −20.32513619513796015329113001370, −19.16462303788284674735714722504, −18.6010356600644153540649353484, −18.039239326569853723137107308777, −17.28525501587597835565936656521, −16.6012331779896366920916584845, −15.65456983548282720173863130485, −14.60605942285022426019945589978, −13.8546577325883090637891498564, −12.95737031628746599221534229984, −12.34764435408678246849958842588, −11.49929383548200949531154474185, −10.703121865656699596445864140779, −9.9277793731684101650648014515, −9.09801832740196557248245710349, −8.725490434604370415353113869624, −7.552831688028124812399037998313, −6.66731745009414522305810254368, −5.50158021765621774892828955473, −4.875992076718475133911387602837, −3.49771001782783280463270146801, −2.57076493629726824208652250253, −2.03027604262753690916291315044, −0.84129690333004445614176784531, 1.13178155350591869503014852858, 1.63095640698093791853335244990, 3.17814093057356320282291033265, 4.40773873477694420703641630444, 5.276587069266941797478402312784, 6.04563713325813596127363665211, 6.80002977252917357592741890741, 7.745089964638925765998404889167, 8.32463238205538138561750183066, 9.51804118393604913003363371984, 9.96659966506000679464049402488, 10.61496231120391714084798326422, 11.630677434873595881715643400276, 12.974448323554968975435069965441, 13.57331775374309860729851023294, 14.36345446599683104914843365006, 14.811156541378931033327464445849, 16.05653376796160520650961578266, 16.62973803558713726293959309907, 17.38372922221341084315410097037, 17.71140090352013802883676488229, 18.77296524342579302914787559369, 19.34740628974678567260335966565, 20.40748571973648404681878087901, 20.950563191529276408557277865939

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