Properties

Degree 1
Conductor $ 2^{3} \cdot 11 $
Sign $0.569 + 0.821i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.569 + 0.821i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.569 + 0.821i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(88\)    =    \(2^{3} \cdot 11\)
\( \varepsilon \)  =  $0.569 + 0.821i$
motivic weight  =  \(0\)
character  :  $\chi_{88} (61, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 88,\ (1:\ ),\ 0.569 + 0.821i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.232335561 + 0.6450670107i$
$L(\frac12,\chi)$  $\approx$  $1.232335561 + 0.6450670107i$
$L(\chi,1)$  $\approx$  1.028713426 + 0.1016977028i
$L(1,\chi)$  $\approx$  1.028713426 + 0.1016977028i

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

−29.77188654631709832276570347871, −29.152781255113473032652641900521, −28.01317487909373380268982531328, −27.05359191855200254736349750405, −26.06195446770086725376437788555, −25.0138243006069176700852521780, −23.65412624951306235216989650168, −22.57582077272919884978617985365, −21.59835428184560916844312418109, −20.57830588648240818483809939108, −19.80515065389516277175814989205, −17.87421498136178311217499701906, −16.95649251934990390112348782009, −16.26542767186745096740827785990, −14.83324053671305937307413826784, −13.67906265640411853563786396046, −12.45013550713133534833999580915, −10.88762888984642201805825962941, −9.91007914763552782672926415456, −9.056536881774534363059302683771, −7.21072323008547708668317052509, −5.578916450247511463162326727604, −4.64329668653635048299251460587, −3.04060684992476374597102236534, −0.68355976790639787757348277675, 1.70475698839991740982734537839, 2.90128860904498229162537857021, 5.406566860439169185608990359201, 6.292099380628553044692621154305, 7.48602371906752423112321099423, 9.030579340671558885610204776429, 10.34362177175656394963662216126, 11.820034065569490462292416970933, 12.68297981908203418226990820360, 13.95151511139219699498741798935, 14.900423032722751676008178984172, 16.6453338573249602574135818028, 17.60289650951808046497799945217, 18.743988862670540747229560819154, 19.20993371363434480920284094879, 21.0533038195957028066779220230, 22.0996456337888445259206122324, 22.95478983418423573152041476025, 24.302573697300442573683614825633, 25.1856696204787385778045399424, 25.91531329339853084146541299860, 27.43098196710261079718438386768, 28.97133122718438979204234273575, 29.10435136831568293095967025920, 30.45339462083408163257944477440

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