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Properties

Label 1-1287-1287.1096-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.996 + 0.0828i$
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  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)5-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.669 − 0.743i)17-s + (0.913 − 0.406i)19-s + (−0.309 − 0.951i)20-s + 23-s + (−0.104 + 0.994i)25-s + (−0.978 + 0.207i)28-s + (0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 − 0.743i)5-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.669 − 0.743i)17-s + (0.913 − 0.406i)19-s + (−0.309 − 0.951i)20-s + 23-s + (−0.104 + 0.994i)25-s + (−0.978 + 0.207i)28-s + (0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.996 + 0.0828i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1096, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.996 + 0.0828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6264809541 + 0.02598768643i\)
\(L(\frac12)\) \(\approx\) \(0.6264809541 + 0.02598768643i\)
\(L(1)\) \(\approx\) \(0.5285505983 - 0.07763989564i\)
\(L(1)\) \(\approx\) \(0.5285505983 - 0.07763989564i\)

Euler product

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

−20.45184904017670853473235089712, −19.86400593591283654745322895339, −19.08064702910038437944106652323, −18.78498592603161144456749049164, −17.70315304561658071189867247412, −17.07884624100309459980292441575, −16.23676615508331615304426216044, −15.54087453080808186982619402045, −15.005783360201636600035924147351, −14.00742012166515688554579194138, −13.09561132654707253445359811755, −11.9393587770808330635242115845, −11.36824276458524253034302666491, −10.38515696043752780820807748531, −10.03974445947770954561058341570, −8.986205567020920463827239573505, −8.086074659903336637935218114254, −7.36925933465564762758912916422, −6.63619871029843121812137980440, −6.058331627431096656541359913563, −4.61968621084313497702439738918, −3.425176237376290040770164183109, −2.83433066949698766361373129197, −1.504142738885404323097898564599, −0.34398992607295462999425187105, 0.50148982540500240339134685455, 1.51995054488783868709701989616, 2.81885930928575023588911129859, 3.39545463301974169139215802937, 4.72602525296366382325442775580, 5.647233654601240285421013602783, 6.89838522291166327728595429158, 7.28694495871359653118094602972, 8.62477760282044795786866071062, 8.855556098544502920227561270618, 9.6837398507831818141869211911, 10.60776124165501996466666687900, 11.62028316009635456857864390635, 12.052839118809422850386495468282, 12.86244820593679560096752139842, 13.69228711125636334876206546101, 15.26025383708669125343795271440, 15.57059928293330866445924861929, 16.33817242060634028803439240021, 16.92173903355836947556873465574, 17.9072090173148953910763516601, 18.6060522625169497918055916876, 19.3485250454420148834694152960, 19.94606881244052477804105310735, 20.53123542435904911135427015458

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