Properties

Label 1-1287-1287.1228-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.618 + 0.785i$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.406 + 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (−0.207 − 0.978i)28-s + (0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.207 + 0.978i)5-s + (0.406 − 0.913i)7-s + (−0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 + 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (0.406 + 0.913i)19-s + (0.743 + 0.669i)20-s + (0.5 − 0.866i)23-s + (−0.913 + 0.406i)25-s + (−0.207 − 0.978i)28-s + (0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.618 + 0.785i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.618 + 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.002616848 + 0.4868836870i\)
\(L(\frac12)\) \(\approx\) \(1.002616848 + 0.4868836870i\)
\(L(1)\) \(\approx\) \(0.7996500995 + 0.1942558512i\)
\(L(1)\) \(\approx\) \(0.7996500995 + 0.1942558512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (0.207 + 0.978i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (0.406 + 0.913i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.743 - 0.669i)T \)
37 \( 1 + (-0.406 + 0.913i)T \)
41 \( 1 + (0.406 + 0.913i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.406 + 0.913i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.743 - 0.669i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−20.9377795477331945409012309324, −19.97453306408023087430729569986, −19.49384191101261285137779386993, −18.490202039458129935310667705346, −17.8538785709379158719713234609, −17.31016865003221781176969667780, −16.26586790324111565201405050069, −15.904123547927352727572695036242, −14.99406523392372798582756632354, −13.888744129869123817078734162817, −12.891463720797296662045018481207, −12.18111132018294504887427513680, −11.62398147844630378892686651050, −10.75441032947313468100498763483, −9.64144243357308348579692807153, −9.12125546957207301364517268143, −8.53155539594760506174243428009, −7.645974188675131701602287578565, −6.79305286187864861830500967086, −5.51530467718414254466612917437, −5.035327856475942462378212754596, −3.61030425622089063960336913957, −2.60456948871084115740763995977, −1.698386822648272190061023092960, −0.76386359940887971859223789248, 0.979927683632109507299924186178, 1.94235529723118567050901083978, 2.99930341711009177940639253971, 3.99324365421282770613296378166, 5.31553400931666550777727831489, 6.26201564760259049938356380240, 6.8885418523975829464620194270, 7.79066311764896057287903701275, 8.23812581718174497805937814887, 9.55766308874595777417389587885, 10.18636551773756772860183118577, 10.7513666822410412582958737960, 11.45005156757578169621729985756, 12.449658715418801401960714474246, 13.74313686027608771783596183467, 14.39756520330387032088272343081, 14.94385450289967984081620034297, 15.84468475242993074176399111441, 16.92015512981494187557539345699, 17.13587212964851642114896822181, 18.17613821802753006099152238183, 18.698199671522536783364387436318, 19.36899352547013157384958628699, 20.3111036187766543037160001732, 20.87273002386958235901978042233

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