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Properties

Label 1-1287-1287.367-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.801 - 0.598i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.669 + 0.743i)5-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)10-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (−0.978 − 0.207i)20-s + (−0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.669 + 0.743i)28-s + (−0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.669 + 0.743i)5-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)10-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (−0.978 − 0.207i)20-s + (−0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.669 + 0.743i)28-s + (−0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.801 - 0.598i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2141398280 + 0.6450524942i\)
\(L(\frac12)\) \(\approx\) \(-0.2141398280 + 0.6450524942i\)
\(L(1)\) \(\approx\) \(0.7507043332 + 0.5899007733i\)
\(L(1)\) \(\approx\) \(0.7507043332 + 0.5899007733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−20.55982883434364228453942383995, −19.94903957623664624634456843623, −19.165729696456469809371543751340, −18.28553454173801390235221247319, −17.72850543954730439788475845222, −16.956080316466550500406990863147, −15.73429264545689729430577911407, −15.18888304349658236637236840750, −14.132386254126096867933471948263, −13.275644990071984722710315816634, −12.96786779111799787445109225759, −11.91730663068944202314583400428, −11.460368280932458869548736583217, −10.38065324139253242011914328984, −9.499380679745503058886036711673, −9.00202000829068744967838415142, −8.34492492625116445598875403910, −6.81231972522080897327317439329, −5.73795570331858348521907437014, −5.24926013199718458110747002367, −4.382152345361214270341096951, −3.2932254552165934863491350544, −2.19893043894077661834189928309, −1.75955683764259721155800583792, −0.22126861341262755810485358157, 1.57724803135605317383178686662, 2.86717032816278155351044134674, 3.83500733178584976740273511205, 4.53056959936185054631539198670, 5.74725711835903933540248415016, 6.31809696894207448382629474481, 7.13424991267865191578966402845, 7.744611363557860250465981515439, 8.82034249326621045346321080348, 9.676517535603650590506986475643, 10.47062082064256907254066184941, 11.22829676336345022711372126489, 12.58069438617799339063735589010, 13.17426023156509542676847316984, 13.95577409036852958977903239716, 14.53817053161313133974517723909, 15.13083504792569712065353728739, 16.3834728932180232532233080954, 16.64493740618050235736656777923, 17.69272176939464022972867455510, 18.10717171229068745700184271118, 18.944214745885346746802239270780, 20.01775259357504031473983244791, 20.85090394688736088404835848496, 21.714550135821546799720818704885

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