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Properties

Label 1-1287-1287.151-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.327 - 0.944i$
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  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.207 − 0.978i)5-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + 10-s + (−0.104 − 0.994i)14-s + (0.669 − 0.743i)16-s + (0.309 + 0.951i)17-s + (0.587 + 0.809i)19-s + (0.207 + 0.978i)20-s + (0.5 − 0.866i)23-s + (−0.913 − 0.406i)25-s + (0.951 − 0.309i)28-s + (0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.207 − 0.978i)5-s + (−0.994 − 0.104i)7-s + (−0.587 − 0.809i)8-s + 10-s + (−0.104 − 0.994i)14-s + (0.669 − 0.743i)16-s + (0.309 + 0.951i)17-s + (0.587 + 0.809i)19-s + (0.207 + 0.978i)20-s + (0.5 − 0.866i)23-s + (−0.913 − 0.406i)25-s + (0.951 − 0.309i)28-s + (0.104 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.327 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5477099560 - 0.3896765792i\)
\(L(\frac12)\) \(\approx\) \(0.5477099560 - 0.3896765792i\)
\(L(1)\) \(\approx\) \(0.8249276246 + 0.1911466265i\)
\(L(1)\) \(\approx\) \(0.8249276246 + 0.1911466265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
5 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 + (-0.994 - 0.104i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
31 \( 1 + (-0.743 + 0.669i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.994 + 0.104i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.406 - 0.913i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.406 - 0.913i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.207 - 0.978i)T \)
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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

−21.292337475806509218762483431099, −20.280311019927538138743892676209, −19.65220208703226214378725796403, −18.94406274047671888663942105436, −18.31610281689750429000687958413, −17.70868564194400444893956446675, −16.63888548514233982591897338310, −15.61450486843190234947780438757, −14.91474580598181601359534014038, −13.924199201483596288814284382466, −13.50847559650852289868302631778, −12.61362912631293947732146168342, −11.7657371110113843687606465482, −11.04248999194105484158296128109, −10.3334052484110746396412154728, −9.43182595940324419973401805081, −9.126802323543460906195652418563, −7.58136527999551792744909706448, −6.839444074925445777937294826133, −5.81254324188752766382736660521, −5.07243438377680356488926469494, −3.784352529688611798171281993306, −3.07579319160949250572990496646, −2.54300013462127590917036286003, −1.25327783653066903381028807017, 0.26153905782840338595713232185, 1.60565628701321630138721608755, 3.21451390893152484051760563999, 3.942151864258663155614210523106, 4.92283175047757819497489713498, 5.71362017949455524305156605912, 6.40007870962403235401829811275, 7.28996480824929581299331366163, 8.317830079240981561416212019797, 8.811912692086319978088168118679, 9.784682229141285027612994813684, 10.32308840176731963282699409265, 12.04153214156825939176378276466, 12.46038886394139226157166569228, 13.31256107303833264657868603118, 13.78740120973644234390135121450, 14.879736473275146743484807377898, 15.55994830058028540628819364039, 16.53443987883534961508372360901, 16.67398523580802263811751436995, 17.518348666996205736135461959996, 18.53205240053736021560809292679, 19.18577723805847986695705186058, 20.15457640482578045190305907706, 20.94749062045916708286094019763

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