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Properties

Label 1-1287-1287.464-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.802 + 0.596i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.913 − 0.406i)5-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)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 + (−0.669 − 0.743i)19-s + (0.104 − 0.994i)20-s + (0.5 − 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.913 − 0.406i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.913 − 0.406i)5-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)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 + (−0.669 − 0.743i)19-s + (0.104 − 0.994i)20-s + (0.5 − 0.866i)23-s + (0.669 + 0.743i)25-s + (−0.913 − 0.406i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.802 + 0.596i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.802 + 0.596i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001815458348 + 0.005485093839i\)
\(L(\frac12)\) \(\approx\) \(0.001815458348 + 0.005485093839i\)
\(L(1)\) \(\approx\) \(0.4752462967 - 0.1166420419i\)
\(L(1)\) \(\approx\) \(0.4752462967 - 0.1166420419i\)

Euler product

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

−21.33455111569549415312133376332, −20.4020805895594012791277353704, −19.58313859229917915822090319694, −19.22299674967453897409805960205, −18.573553154070261590497498275558, −17.48318570947204266770975282126, −16.887604772900679388955483905215, −16.21768015435508449095711464817, −15.335560456992838770630030518753, −14.93539117822120042139919497889, −13.94211889589346536977599612527, −13.06389223170803356957426814751, −12.00881424987219780225734886788, −11.083397717092490341768521061307, −10.474877596451350849266816321414, −9.76257655349190976493921324752, −8.776609023797481230407131707, −7.870000558443779205212313705658, −7.41177899909713010471239810701, −6.45748630763417699583097722227, −5.87358211293101566890262070351, −4.42982431680899131053530479086, −3.739920466887184866199260229638, −2.54357663618633480249841584700, −1.23501342841159056505498189199, 0.003623487993465867763554301564, 1.13756147521560424770705514803, 2.56864183636525033799074206462, 3.08470330382977207985187418525, 4.21057192608424037211330456479, 5.054999488309274330116788712326, 6.49381563942767211228415276775, 7.157829900705122266211735701923, 8.12070350743131063532722206339, 8.95105161206379760364623129595, 9.27213608504348127413564935573, 10.46591361197212943072781596575, 11.21372940699085310629827895697, 11.93402758329793545133604462725, 12.66809836205140710959368486169, 13.11590299580169545462138231192, 14.55062185675861070324055774277, 15.48732253230589180626511377162, 16.1943230492942955326869753568, 16.54588618153620455776323103198, 17.72286507738280852275141332364, 18.40152636028959064824471443906, 19.0996674387490499115000134092, 19.77630243327846157588483767822, 20.22872550027783725868123321521

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