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Properties

Label 1-1287-1287.1022-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.393 + 0.919i$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s − i·8-s + 10-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − i·19-s + (−0.866 − 0.5i)20-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·28-s + (−0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s − i·8-s + 10-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − i·19-s + (−0.866 − 0.5i)20-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·28-s + (−0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.393 + 0.919i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.393 + 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04963321225 - 0.07520838720i\)
\(L(\frac12)\) \(\approx\) \(-0.04963321225 - 0.07520838720i\)
\(L(1)\) \(\approx\) \(0.4538370629 - 0.1426783436i\)
\(L(1)\) \(\approx\) \(0.4538370629 - 0.1426783436i\)

Euler product

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

−21.08830499087032962244761136921, −20.31680521457156927728132282060, −19.54655387013642971403825619128, −19.14620087762585800851064547611, −18.36317855154041752191793349867, −17.46092999822368716307353556694, −16.683534646974199566668349557780, −15.929098647298421365544671068514, −15.55788893253456976626897512522, −14.79569631158886345286136001733, −13.66946383987992300808911474530, −12.77101733132406852115445676084, −11.828765344953625665199024461188, −11.30555744981367348415118388087, −10.10818352322487575805197250285, −9.53879146742587661145037776841, −8.61356776279507203347709827118, −8.09265939019964231306400408139, −7.159292724326553279902803491305, −6.32562946751350600464647863193, −5.5500752391764333221888298549, −4.48594799219703032196912353483, −3.422059450997690876715500818915, −2.273697752164416097503499774875, −1.12287197751570605065861405180, 0.04282347958591395001258367109, 0.60870619141522719259002420965, 2.16157692732251234641843450649, 2.99850840020827046102140997713, 3.79946969476050343443945832093, 4.59764354285000358607827337536, 6.39682411970488538060155625633, 6.84743540183490457701943341865, 7.66487500199593534939301965506, 8.51451871481131128844507564648, 9.34197206007556873180807101046, 10.16100541494302436175519917647, 10.99661609275031150562003202893, 11.39451067853022045818194814707, 12.530924128854135398695062748740, 13.001581394154825668513575175308, 14.09748457328389952088311512235, 15.21054592350509098897823206779, 15.93053228153353669966979592665, 16.40394468652843775274709252343, 17.4108659827740660533454203973, 18.10724508650093799131736522243, 18.903486304305556759843188992610, 19.612736420514293500179722169147, 19.96441762032844542688561576530

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