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Properties

Label 1-1287-1287.1042-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.0785 + 0.996i$
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.207 + 0.978i)5-s + (−0.406 − 0.913i)7-s + (0.587 + 0.809i)8-s + (−0.5 + 0.866i)10-s + (−0.104 − 0.994i)14-s + (0.309 + 0.951i)16-s + (0.669 − 0.743i)17-s + (−0.406 + 0.913i)19-s + (−0.743 + 0.669i)20-s + (0.5 + 0.866i)23-s + (−0.913 − 0.406i)25-s + (0.207 − 0.978i)28-s + (0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.207 + 0.978i)5-s + (−0.406 − 0.913i)7-s + (0.587 + 0.809i)8-s + (−0.5 + 0.866i)10-s + (−0.104 − 0.994i)14-s + (0.309 + 0.951i)16-s + (0.669 − 0.743i)17-s + (−0.406 + 0.913i)19-s + (−0.743 + 0.669i)20-s + (0.5 + 0.866i)23-s + (−0.913 − 0.406i)25-s + (0.207 − 0.978i)28-s + (0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0785 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0785 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.0785 + 0.996i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.0785 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.763419186 + 1.907867082i\)
\(L(\frac12)\) \(\approx\) \(1.763419186 + 1.907867082i\)
\(L(1)\) \(\approx\) \(1.613946145 + 0.7229456500i\)
\(L(1)\) \(\approx\) \(1.613946145 + 0.7229456500i\)

Euler product

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

−21.033633635181906998896440310138, −20.13861847517830053169932297698, −19.360604617496041999125641209625, −18.964467094193392627380760820194, −17.70077228441920803129816885313, −16.73141115780419393367655628289, −16.06445664700512324829402403132, −15.33239479084742735365287312773, −14.77500477763725755958302865790, −13.62614400461221053221287619204, −13.035139442126636403513046520536, −12.16711003258431759780611188448, −11.99898575073885681069816746588, −10.78385531507180363174499995856, −9.98615147398436294021720208887, −8.95718883189780790360973454750, −8.327592811619024254005482530242, −7.07679547772422763827800714520, −6.183452458250162177183295610750, −5.420778381325244490424785206112, −4.7032957806601746414238953436, −3.83728912528772757466901329230, −2.81549021015591114819325560391, −1.96667652095471773160886061063, −0.75589966710493656747443780056, 1.3737198496215538506707168727, 2.783604078189375365209701451482, 3.32112863884368372260546660966, 4.15229169412788523476848324303, 5.08614186865385824377554170290, 6.22940578663530648147783651099, 6.72575688168144767020361767882, 7.57769673879300924633089534163, 8.14662091148878206492596392083, 9.78544329436602635390700012986, 10.341700030663926334752062049978, 11.33826012412175028006567463120, 11.82718959384581410550859318828, 12.957076540614720455197963771763, 13.56150256935553096534962373871, 14.380790442507382679411324362634, 14.82126783626454786349569993340, 15.81921262499357951210267642942, 16.42973303932161612858915216922, 17.20648095274612621139203864826, 18.09619365695821125175940759410, 19.13620041183984576213682559730, 19.69692670014541158380852529103, 20.68126248312259017651260707998, 21.226202062409412999081494289

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