Properties

Label 1-1287-1287.149-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.776 + 0.629i$
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 + (−0.994 − 0.104i)5-s + (0.207 + 0.978i)7-s + (−0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.809 − 0.587i)16-s + (−0.913 + 0.406i)17-s + (0.207 − 0.978i)19-s + (0.406 − 0.913i)20-s + (−0.5 + 0.866i)23-s + (0.978 + 0.207i)25-s + (−0.994 − 0.104i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (−0.994 − 0.104i)5-s + (0.207 + 0.978i)7-s + (−0.951 + 0.309i)8-s + (−0.5 − 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.809 − 0.587i)16-s + (−0.913 + 0.406i)17-s + (0.207 − 0.978i)19-s + (0.406 − 0.913i)20-s + (−0.5 + 0.866i)23-s + (0.978 + 0.207i)25-s + (−0.994 − 0.104i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.776 + 0.629i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.776 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311045961 + 0.4648746144i\)
\(L(\frac12)\) \(\approx\) \(1.311045961 + 0.4648746144i\)
\(L(1)\) \(\approx\) \(0.8797822220 + 0.5367540018i\)
\(L(1)\) \(\approx\) \(0.8797822220 + 0.5367540018i\)

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

−20.53582328819129238488113016230, −20.05180273894991433174441742363, −19.551192522946592012438289010659, −18.53826147639953849209036769203, −18.03489099335928870291109384621, −16.80977034071712667056946005828, −16.02996588795109811143985966408, −15.224275205345084132114011606567, −14.37699764626794701339427374704, −13.83539543793242339595284748009, −12.918820311219735288505470259279, −12.08242991400846655986147182119, −11.54339825798918888250854149993, −10.526339856984552429300395489921, −10.30344045822590076429568709233, −8.95257159126779849946547121351, −8.17411976977905591141086565018, −7.09626295290315448180569360241, −6.420423417717689182318824550786, −5.06220655846420725118018067266, −4.42257850668607775742704719415, −3.68578110560293461669729314092, −2.90485207856125889183026942580, −1.63791336299828263968519325648, −0.66069808630457660277480176531, 0.33514269434772893518192592474, 2.15825686200888026344401643559, 3.10672524943198551097797158771, 4.096673721140284558796536449396, 4.76629368365989245602221970531, 5.67449312487550233177711341318, 6.516666310643776792883309316919, 7.41336649230642439249729578508, 8.19635149538219941725477850192, 8.77246073891857103755562362649, 9.6593312003993999686422544602, 11.32362997557464255687976949743, 11.55035277446293679026177454798, 12.51722727105694883843006501610, 13.16336313475333270061122625921, 14.08485313456667508025529867467, 15.045645710991825477516909600211, 15.54992884624918342971558679936, 15.8901729285579715342119874711, 17.028545162195459199306535928458, 17.71201340417825097476267205778, 18.50061203150195091913360001129, 19.38801718023708233782735441849, 20.15073663679643192115730898174, 21.14355853298129193102363466136

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