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Properties

Label 1-1287-1287.1091-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.751 + 0.659i$
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 + (−0.104 − 0.994i)5-s + (0.669 − 0.743i)7-s + (−0.309 + 0.951i)8-s − 10-s + (−0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.104 + 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.809 + 0.587i)28-s + (0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (0.669 − 0.743i)7-s + (−0.309 + 0.951i)8-s − 10-s + (−0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.104 + 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.809 + 0.587i)28-s + (0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3570329264 - 0.9483705198i\)
\(L(\frac12)\) \(\approx\) \(-0.3570329264 - 0.9483705198i\)
\(L(1)\) \(\approx\) \(0.5868075690 - 0.7357073129i\)
\(L(1)\) \(\approx\) \(0.5868075690 - 0.7357073129i\)

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 \( 1 + (-0.104 - 0.994i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−21.75318858648022097589551335119, −20.97062450914150411989336621191, −19.79860324907528309868119316470, −18.81176631097827300753732393649, −18.2672770165936339660105995758, −17.81425437780604620140234126172, −16.869661059255109010764780578964, −15.94099405788839718053747862551, −15.24389910292817170604056564393, −14.74863432047183760563511967908, −13.996137499664270262342843470899, −13.29718900588390960299719940867, −12.14985866353367657049484409185, −11.48336829476162741054152394173, −10.49612326623791897745293234048, −9.55311136154198361728642652451, −8.74240706877679907112251172583, −7.908234934028698894751375536443, −7.18367717855416742399674934614, −6.41324444633392263912262248288, −5.54054112589219492087563548763, −4.81966924402914667145142983360, −3.69972532401138268996064987488, −2.84785675601882067942139934210, −1.58746832575527290656406533333, 0.40079151879089701247648027538, 1.37224897536618507426654753882, 2.238614374028827767203379527374, 3.47567991127912926874262724099, 4.49970711246662574742484508456, 4.74294500887995348861338981480, 5.85368558071902705603320777141, 7.15885784244202866757184161052, 8.21128183876452849567964333500, 8.80334942089464773950245071936, 9.56481971828853725617891310536, 10.641929659380809819616304815497, 11.10827259083414485149127875112, 12.01792732593430630780749842411, 12.7841468615980681646568213416, 13.41140400905158956166723242957, 14.11426969745544015731499509481, 15.019269564718629547662947289167, 16.002528391786599308314367288558, 17.05398071396776329241392023256, 17.48843987492558277351542368607, 18.24764398628129192763905227626, 19.35788462315833087707339449806, 19.9603964357064260830825037101, 20.40575598227653316070328521395

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