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Properties

Label 1-88-88.13-r1-0-0
Degree $1$
Conductor $88$
Sign $0.569 - 0.821i$
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  + (−0.309 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(9.45691\)
Root analytic conductor: \(9.45691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (1:\ ),\ 0.569 - 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232335561 - 0.6450670107i\)
\(L(\frac12)\) \(\approx\) \(1.232335561 - 0.6450670107i\)
\(L(1)\) \(\approx\) \(1.028713426 - 0.1016977028i\)
\(L(1)\) \(\approx\) \(1.028713426 - 0.1016977028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

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

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