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Properties

Label 1-1287-1287.511-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.867 + 0.498i$
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  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.913 + 0.406i)5-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.5 − 0.866i)10-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (0.978 + 0.207i)19-s + (0.809 − 0.587i)20-s + 23-s + (0.669 − 0.743i)25-s + (0.104 − 0.994i)28-s + (0.669 + 0.743i)29-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.913 + 0.406i)5-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.5 − 0.866i)10-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (0.978 + 0.207i)19-s + (0.809 − 0.587i)20-s + 23-s + (0.669 − 0.743i)25-s + (0.104 − 0.994i)28-s + (0.669 + 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.867 + 0.498i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.867 + 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2933012989 + 1.098945458i\)
\(L(\frac12)\) \(\approx\) \(0.2933012989 + 1.098945458i\)
\(L(1)\) \(\approx\) \(0.6696000952 + 0.6054552896i\)
\(L(1)\) \(\approx\) \(0.6696000952 + 0.6054552896i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (-0.913 + 0.406i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.978 - 0.207i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.913 + 0.406i)T \)
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 \( 1 + (0.809 - 0.587i)T \)
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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

−20.714464359830029912920816812635, −19.84861883819688362399485518089, −19.41179732322101488498269702405, −18.71812004592229139907618121245, −17.739546062649006632529266295647, −16.866939631975346497083085661557, −16.2950334690528906933243997691, −15.154958070832862438898325466874, −14.45500181314188230259762617002, −13.39336043412834505763885267945, −13.01965162951435474624848745569, −11.98015639005782313170198939789, −11.49103781007006841074449293260, −10.617501914708548469751577604870, −9.84334438291669689313290294985, −9.10102091564046848837543165674, −8.00144309146885721301413780355, −7.52333440108238680072485679248, −6.21088551513342358795252028433, −5.03227716317928908229749235703, −4.34652744017131152776951489782, −3.533045395911440812782290939622, −2.863841137339344086873298085460, −1.31834269087394118179962772920, −0.61620437224985911444796194024, 1.00775789455930083140538941316, 2.92658581215919564111183848083, 3.377438506546772439469364077315, 4.61711303596843686883156836703, 5.349217903300791081153100368123, 6.22796802736198711208664542817, 7.13391969324835936918625857435, 7.72242573215335919908061718628, 8.63418450878328266274858037831, 9.29816300091867112777157524106, 10.2431750141248653454282729629, 11.37615851741110647371727048124, 12.26062829978575385706691267922, 12.70379477063626462235300803675, 13.961885663988343850360962383500, 14.54976206292839662184547702428, 15.26999240482230227829427222214, 16.05503545986959017201634694644, 16.30998072233471642252018516819, 17.55261686935102659206609363228, 18.28453842759062452702815772805, 18.90612089766045358699682912096, 19.46325925850231543057324108608, 20.64304639383739073368528332724, 21.5994012171073113115520545756

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