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Properties

Label 1-1287-1287.137-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.487 - 0.873i$
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  + (−0.994 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.487 - 0.873i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.487 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3262980007 - 0.5555411731i\)
\(L(\frac12)\) \(\approx\) \(0.3262980007 - 0.5555411731i\)
\(L(1)\) \(\approx\) \(0.5735189045 - 0.1844123463i\)
\(L(1)\) \(\approx\) \(0.5735189045 - 0.1844123463i\)

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 \( 1 + (-0.587 - 0.809i)T \)
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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

−21.26060315774171139889319700540, −20.23211960379666350017085585426, −19.49310973756997243788321823967, −18.86666764927566643074274605756, −18.502516862658359084089436257026, −17.57483444616823498998285205002, −16.55105326925632823864125438799, −16.19271887797031227682703525933, −15.09921994083413073399915570795, −14.77621909867119457303476189298, −13.51099414800262224660344402956, −12.36817755251801895083869948141, −11.96850705967975996434915549149, −10.86032975805309475213921099643, −10.33778456082957868490137618284, −9.5756989133787229126926082997, −8.752008489798701729105057130126, −7.75716073662544108911781837999, −7.16068236840191422681668469730, −6.3013052061441401406517742877, −5.61405711413582365536076121361, −3.83540034462269978755268096365, −3.17380765035512768922028399584, −2.40818227241525760793461308456, −1.10864566225474358722380853701, 0.44075786932864121368765468131, 1.22572654488760305502029244255, 2.64198718174155048407378179690, 3.4609012585270937175857430825, 4.69171396001370230252708655619, 5.664933995036271721582171078873, 6.61781628839394908817182552143, 7.42892933051163892145309346371, 8.12775392505676507903500865033, 9.14743126533426687464187630614, 9.52678205311138513264897941867, 10.41805635466415581982993569669, 11.4208558353417104254319478908, 12.060119536945023080133023277430, 12.93717230560245820435207565659, 13.67855010285075500649525663183, 15.00005785413026782496979969462, 15.64250725426482615878084823145, 16.324245859186728896675108681028, 16.932600317875648669230679042156, 17.49643990473976368481396721761, 18.68906790929577841262212886364, 19.211452265120168870553557146027, 19.8358141702984225964553864420, 20.60917553799173627741890935645

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