Properties

Label 1-1287-1287.1021-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.0366 - 0.999i$
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  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.406 + 0.913i)5-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)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 + (−0.743 + 0.669i)19-s + (0.994 + 0.104i)20-s + (0.5 − 0.866i)23-s + (−0.669 − 0.743i)25-s + (−0.406 + 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.406 + 0.913i)5-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)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 + (−0.743 + 0.669i)19-s + (0.994 + 0.104i)20-s + (0.5 − 0.866i)23-s + (−0.669 − 0.743i)25-s + (−0.406 + 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0366 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0366 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.0366 - 0.999i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.0366 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063764594 - 1.025482060i\)
\(L(\frac12)\) \(\approx\) \(1.063764594 - 1.025482060i\)
\(L(1)\) \(\approx\) \(0.9502654786 - 0.4071766053i\)
\(L(1)\) \(\approx\) \(0.9502654786 - 0.4071766053i\)

Euler product

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

−21.04710988084550094892699195577, −20.48009214877878633206223804017, −19.40864856683111098834872514355, −18.77218680833254729377791411091, −17.66349034793124055703575904912, −17.00735553382127953341340422650, −16.267353723530359151050227098600, −15.50603770496211721410503760760, −15.2782534483585968669714195168, −13.95091413763414294868942371930, −13.33297882908363245481388924582, −12.56875964914179825751545115033, −12.02878759364322144840101879354, −11.19641462300051327132527230553, −9.654330385087196701219002001904, −9.032748182891069457166422098344, −8.39933632819120753221130516911, −7.38928622053941031624764750032, −6.68740268919318117596211288039, −5.604797256044205070025019230734, −5.10672333908411469847029883119, −4.12538462977804092993227116728, −3.28099386543143623270490776963, −2.2632892526807249357298768128, −0.56817379290075541953317546363, 0.39920937718230446024127461757, 1.694723177369919150289678110115, 2.71701100545890637563501054971, 3.63527737855006101553701370667, 4.00789436798076226169546754741, 5.236468776351814106668894110296, 6.38340091152586351233707556930, 6.75128837464135638140199207561, 7.96194348035261437979034093726, 9.02544149212411630373925139940, 10.07467976009511116414543574809, 10.6160466351617344603182304000, 11.08485808111278840425548552490, 12.28966990358554278660318599217, 12.74589863281441853324646519746, 13.625252555526322305281295991399, 14.56806848972274277786049134481, 14.85022524639297536863221032051, 15.92700669313030732892492981083, 16.74496219470515478060467279765, 17.83442068113604506126766516859, 18.65947443556026179102877427143, 19.285414056703468993399100425649, 19.76639939217088340372616298929, 20.63142504352797349115206856353

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