Properties

Label 1-1287-1287.860-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.991 + 0.126i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (−0.406 + 0.913i)5-s + (0.951 − 0.309i)7-s + (0.951 + 0.309i)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 + (−0.207 − 0.978i)19-s + (−0.587 + 0.809i)20-s + 23-s + (−0.669 − 0.743i)25-s + (0.994 − 0.104i)28-s + (0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.978 + 0.207i)4-s + (−0.406 + 0.913i)5-s + (0.951 − 0.309i)7-s + (0.951 + 0.309i)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 + (−0.207 − 0.978i)19-s + (−0.587 + 0.809i)20-s + 23-s + (−0.669 − 0.743i)25-s + (0.994 − 0.104i)28-s + (0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (860, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.119768448 + 0.3249187923i\)
\(L(\frac12)\) \(\approx\) \(5.119768448 + 0.3249187923i\)
\(L(1)\) \(\approx\) \(2.204381919 + 0.2538423025i\)
\(L(1)\) \(\approx\) \(2.204381919 + 0.2538423025i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−20.815731911779801976872988710405, −20.36366653634472073743629290657, −19.4781419075513045482173796276, −18.752421581069096720003716781211, −17.51530473349943115531190717407, −16.90629236291675540515306682945, −15.99943514713519326443413622761, −15.32425076845824881987861692378, −14.69083872074576679873155883881, −13.81497042368391875085043968984, −13.01290606975717292595543815261, −12.32832849091441442669593084297, −11.65343736968769948596750150604, −10.982179909681397343128738913120, −10.04363651372086905562181198219, −8.73864211912715720988792947198, −8.204039775321561075310023258246, −7.23827572475713102450193929357, −6.22417474636780399912052084076, −5.30209586906032676060338068550, −4.649942961391668249472798487043, −4.02500447306592079085853059113, −2.85301148737670129232032606971, −1.77240651929422495721560062322, −1.009816725620226463874471547458, 0.78652874958147905267855760906, 2.24294734567915196511178748860, 2.78119286751442649656557455467, 3.98743849081773827045373226194, 4.55705036909136854504466962900, 5.46617988654664660387048064692, 6.62229431873160141985794948480, 7.0959356325322905352733042811, 7.90295838953884741423679385980, 8.86104450593819679113179294194, 10.30957005186905212797115999807, 10.965970290527951088382267656, 11.47762461086453894798186381111, 12.22340291328216668034084019832, 13.420131152311007730296759547306, 13.857802008819717709993822333141, 14.70016003009277299583108115851, 15.3540258608177184752951415616, 15.81782224027610237774830717330, 17.09974495575772079501178756100, 17.61102057631383617982670262859, 18.610137137786677958697127138228, 19.55675740453318111211828677633, 20.105713580004608961324207813726, 21.09518984212518261835409537342

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