↑

Properties

Label 1-280-280.179-r1-0-0
Degree $1$
Conductor $280$
Sign $-0.832 + 0.553i$
Analytic cond. $30.0901$
Root an. cond. $30.0901$
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  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s − 27-s − 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + 41-s − 43-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s − 27-s − 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (0.5 + 0.866i)39-s + 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(30.0901\)
Root analytic conductor: \(30.0901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (1:\ ),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4428338173 + 1.465297544i\)
\(L(\frac12)\) \(\approx\) \(0.4428338173 + 1.465297544i\)
\(L(1)\) \(\approx\) \(1.009270180 + 0.5003227362i\)
\(L(1)\) \(\approx\) \(1.009270180 + 0.5003227362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - T \)
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

−25.05021989370254767146016666994, −24.17483572346723576406694463547, −23.27671950203905045761441102810, −22.61212237000583426087307802110, −21.06752120155694091818210740665, −20.47768024941720236335654582743, −19.559802922869847602448628599556, −18.449823963409264916038029052837, −18.05153099773099939892455360165, −16.85307416617940085774709542399, −15.63173540248904920573902175400, −14.75314241889424841320253836571, −13.70597152567613952753925980237, −12.97586847683407232098080027342, −12.05903073995689217963321164424, −10.97778430159034529286397752663, −9.67054116060096124640410066852, −8.677643676685375175371632591315, −7.68529071368261922797670933094, −6.82285314544097662039734838817, −5.70736958720785064219983618532, −4.25866909351102841216244673370, −2.89458490039704998736905176248, −1.868758124796610031555778164885, −0.42127419934905526199582660046, 1.63188439048882965749787908944, 3.2120787382510793594903333890, 3.888780030051066402071014386388, 5.29868670964499549338297004884, 6.196477625045303210213394272777, 7.98537605982927102732798126600, 8.4816908984857320083274890470, 9.71579993704247994867191265137, 10.60256207349578787006737052562, 11.402896576436310953506595491265, 12.87541476316043716772366322048, 13.81532536847857179114590303762, 14.65931021370074752304279509812, 15.71156517930979334200285709145, 16.32796886596061618653292365572, 17.342377278815502046009616102530, 18.6956002625239519143243889431, 19.36267054545749511124354439471, 20.54764062646027063568100550406, 21.19079425992292386745115391338, 21.89780002132316834859633551103, 23.04220095077636256458600828743, 23.88141485971783841812855944353, 25.09560609477954520628811044151, 25.89702781747532903376921962519

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