Properties

Degree 1
Conductor $ 2^{3} \cdot 11 $
Sign $0.0694 - 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + 21-s + 23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)35-s + ⋯
L(s,χ)  = 1  + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + 21-s + 23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)35-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0694 - 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0694 - 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(88\)    =    \(2^{3} \cdot 11\)
\( \varepsilon \)  =  $0.0694 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{88} (85, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 88,\ (1:\ ),\ 0.0694 - 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.580318367 - 1.474083857i$
$L(\frac12,\chi)$  $\approx$  $1.580318367 - 1.474083857i$
$L(\chi,1)$  $\approx$  1.317472615 - 0.5796557962i
$L(1,\chi)$  $\approx$  1.317472615 - 0.5796557962i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−30.70764758305521639269301080337, −29.89124628240640129722745656848, −28.22394064239984697179977734307, −27.12606578353562608443555338252, −26.43590569498789163683684163671, −25.63873231144814621780607906720, −24.184757552436156902991875668030, −23.16966831751842886669116512513, −21.75902671501372864203108855126, −21.09272215203644756976416123375, −19.7611439035145771612080427031, −18.98032410265048790926183684195, −17.60333717903255964934541134332, −16.26476708888250951727460058445, −14.92699909733810538593557986789, −14.43827092721007228972773988722, −13.206074311300094959890273849498, −11.19197271954793808878118762330, −10.61523850568716937692503742896, −9.07948157394368220270860178363, −7.901952890187410408153457226724, −6.723514376525994653766760561067, −4.620301107031937831681107899738, −3.57917322839165950955488910064, −2.01997402616371805527573656133, 0.99093377377443232274418577376, 2.52098203463026071774356728765, 4.26884258444892216445313669469, 5.74134216253570842497042382103, 7.5967806158838938743872408620, 8.4193877454491709847254509102, 9.38976415642419695316281542646, 11.337640587836503163328514329151, 12.51618494294723308682024456419, 13.365157915124857942863865755586, 14.76245578902904146736250483833, 15.62391708278679263277468641317, 17.18326989762809274279412538602, 18.300455600122979415022493666, 19.330719337740095810217628694, 20.605846663563806459515885937480, 20.963197476322305619788881965038, 22.81037463027667711464339261578, 24.05609999744800856290893592026, 24.76852125223016017378230711999, 25.50889565993174422085540504284, 27.081156831476464501903773727776, 27.82140129429963936552214793030, 29.117269244800208486099131161602, 30.21837611020065869359516631953

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