↑

Properties

Label 1-1287-1287.862-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.990 - 0.137i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.990 - 0.137i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (862, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.990 - 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.063032226 - 0.2115857558i\)
\(L(\frac12)\) \(\approx\) \(3.063032226 - 0.2115857558i\)
\(L(1)\) \(\approx\) \(1.972095213 + 0.01433655710i\)
\(L(1)\) \(\approx\) \(1.972095213 + 0.01433655710i\)

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 \( 1 + (-0.309 - 0.951i)T \)
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

−21.04914813880082097666034653215, −20.57567298366120551178856952185, −19.34699687113177665477933422825, −19.07327772355238801250564961036, −18.12863155221359555796015370049, −17.17416730143545836046359863990, −16.073072947138751710535767552066, −15.46633656506128624789771424790, −14.720788853744692414728186044247, −14.324238231486323869017973302099, −13.3606893113270871839496614963, −12.30684880184028909054045389951, −11.86800467509610095715534577214, −10.95475436736908207075408732607, −10.604116590370532250046192952511, −9.29388597877965850365901351519, −8.21304068398184800715434250500, −7.34943448482833650415522380106, −6.67955938814997375016836327784, −5.6505375541759401930428157969, −4.843366108844576702663803112041, −4.07106873201237136596819180488, −2.982968234751204876389887595779, −2.44742898765689653418796187507, −1.16455581834783054517984454738, 1.05668779378286661861307615364, 2.037477730967191183485715648264, 3.36134047706148698941081516376, 4.16944496983838231046957850207, 4.71649530076526829241059820733, 5.60150151531090586969661579089, 6.55903682602623030166999185078, 7.666721374619329022429249135032, 8.004916264506865862523488487654, 8.9698527133281784454105026000, 10.42421425289931823069163190697, 10.99881168218565086509291782097, 11.93046444777518747363844106278, 12.466157235476827630502623306340, 13.29385478423452733847506842484, 14.04607240728481012956287104478, 14.94200781082157028421605627610, 15.371352298272613955933409101811, 16.432294368981461016830390754490, 16.97800541031619203181028633386, 17.55707689538749198776698789741, 19.094835737714330295994065107872, 19.53418609953264303557202969627, 20.58768911246628266339550028689, 20.97713479654923665924818294299

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