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Properties

Label 1-88-88.85-r1-0-0
Degree $1$
Conductor $88$
Sign $0.0694 - 0.997i$
Analytic cond. $9.45691$
Root an. cond. $9.45691$
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  + (0.809 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(9.45691\)
Root analytic conductor: \(9.45691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (1:\ ),\ 0.0694 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.580318367 - 1.474083857i\)
\(L(\frac12)\) \(\approx\) \(1.580318367 - 1.474083857i\)
\(L(1)\) \(\approx\) \(1.317472615 - 0.5796557962i\)
\(L(1)\) \(\approx\) \(1.317472615 - 0.5796557962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + T \)
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

−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