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Properties

Label 1-1287-1287.1285-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.955 - 0.295i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.955 - 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.965988208 - 0.4476571231i\)
\(L(\frac12)\) \(\approx\) \(2.965988208 - 0.4476571231i\)
\(L(1)\) \(\approx\) \(1.446431946 - 0.5255128274i\)
\(L(1)\) \(\approx\) \(1.446431946 - 0.5255128274i\)

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

−20.96775755615339194318861896664, −20.424666886500225502034772841426, −19.29295006572344997380330783054, −18.28361094585766615899438771678, −17.56347105305374849285516308694, −16.925338946267859437424147679616, −16.43656631052298817193879244888, −15.40780043512010935469295457679, −14.54377860944625641667880675341, −14.02491341523274198731605800957, −13.1309599087868924633132955257, −12.87810082548448920230306744052, −11.628891223579450704229925045736, −10.61739363071130186273611695654, −9.874844030419046271602884053699, −8.942750509215078354977546475894, −8.06653245256204969291687523102, −7.16284772892874906846164098582, −6.34917503612754605500430373035, −5.89076683584473754785101649403, −4.674081105344417918368301885494, −4.08648703502172685363703635319, −2.977462732075538000145535300107, −1.982764713540207622659637069554, −0.531768024199485333223544455401, 0.906886887785797762502921375723, 1.99541553607721598740218356065, 2.61475338827181192307855214656, 3.48001004746490446734807128785, 4.98912890604090042075185631801, 5.1336022569541861292014361339, 6.29677529035345532836645291331, 6.85769718473901647611755761828, 8.653883681625711582737535443062, 9.11984969406309633207562186767, 9.83131317756465483992799904789, 10.77513677953979381781808819821, 11.45229984707103920030922416326, 12.421492355631758373650673134125, 12.9697568484363492554033383103, 13.74806304754040776150281698034, 14.36095860918278599457532205710, 15.43629150454444140167880993619, 15.814346941718178491214319829101, 17.30252483969046473040188327539, 17.817555640199862797517084673820, 18.62393532488949741338237506763, 19.32498124247849921313066870224, 20.15064379490653399930393465566, 20.92273818363747819099700928011

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