↑

Properties

Label 1-1287-1287.113-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.598 - 0.801i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (−0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + (0.5 − 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.669 − 0.743i)28-s + (0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (−0.104 + 0.994i)7-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + (0.5 − 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.669 − 0.743i)28-s + (0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.598 - 0.801i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.598 - 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3291445134 + 0.6562778517i\)
\(L(\frac12)\) \(\approx\) \(-0.3291445134 + 0.6562778517i\)
\(L(1)\) \(\approx\) \(0.5418195576 + 0.4788393177i\)
\(L(1)\) \(\approx\) \(0.5418195576 + 0.4788393177i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
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 \( 1 + (-0.978 + 0.207i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.42728292125068474274997030689, −19.418465527957646966229069674486, −19.24765643732141952527735944564, −18.112588511281804019136268709287, −17.282264733472957023905091759518, −16.59728065358836948761925393902, −16.10010067403784048419068866863, −14.779930858388855260151324954022, −13.944870664251479626568059983978, −13.152093213248375124521557228439, −12.48166451553685495325283275066, −11.754494417958334379539267818, −11.018640417166162893973733847263, −10.13098717018531544001419440678, −9.50682458836214989611296470296, −8.521512700743341771647203842377, −7.789456900234730833855353985463, −7.18442384361931264829318585596, −5.593817299325017026857495745091, −4.740939988600934126135001305085, −3.76865739176843812896331962846, −3.42030818951897325688009548011, −1.89232847005379918044082628011, −0.98633526147880701132991467834, −0.21867653230307597493132931865, 1.05504922342654064701593563385, 2.54711834723440262301439626836, 3.44184324596464893139869904308, 4.61102918062769586334164115869, 5.38474012608392562643476195171, 6.39587123714503041797825105870, 6.9319633109293781651573626934, 7.90778503049621145057808066447, 8.5602053001357600691438578783, 9.354501009167827731775275134087, 10.30558749843766507384725338078, 11.06245687830793187332473140714, 12.08307300556549867313974012207, 12.82964299366528763216871009132, 13.91156837526272588896180184022, 14.78450021122608901014010031831, 15.08163981280713708495826622084, 15.95370654284967398409087004123, 16.52822005445210087928972655333, 17.59099996528308862115486335292, 18.28556208692079793313598839896, 18.9045171476388253903637338772, 19.38286470168489159374796627991, 20.38844564220214182896112880986, 21.74391852853939758027038772477

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