Properties

Label 1-88-88.27-r1-0-0
Degree $1$
Conductor $88$
Sign $-0.550 + 0.835i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.550 + 0.835i$
Analytic conductor: \(9.45691\)
Root analytic conductor: \(9.45691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (1:\ ),\ -0.550 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8717271307 + 1.618234264i\)
\(L(\frac12)\) \(\approx\) \(0.8717271307 + 1.618234264i\)
\(L(1)\) \(\approx\) \(1.063876633 + 0.7009158734i\)
\(L(1)\) \(\approx\) \(1.063876633 + 0.7009158734i\)

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

−30.00192904865595353401837890430, −28.963869678393344678479471130254, −28.269885141424961700440537630318, −26.3192758954459306900119800905, −25.85889146480390728395904278025, −24.49387991526061260522257230779, −23.89624808248311798151638370917, −22.68286523707072947489309241888, −21.19973868433955540980026946276, −20.18475133814668525642067153550, −19.31399936838296020249616271329, −17.8815935218753706260705931206, −17.22103884441245449237344502630, −15.88730946534182392555298804465, −14.011155717705988705370987681255, −13.53303888003004614791877985027, −12.50669915518956772334031299391, −11.00153350922352094533657185925, −9.481510806319495884864674875994, −8.39268345715794398933695970449, −6.95131175648501722355337973554, −5.95746989830095033570418727317, −4.083778619882557491742377805612, −2.20407205182165130142839466096, −0.83183133692998259340340397727, 2.306300389137098560731572153555, 3.493119441699387204655774270311, 5.291098322690379749205319812038, 6.28991413758862863630228197953, 8.29064239039455006468524950596, 9.439070044473642036356057784760, 10.33152687561207446399690741185, 11.56600983106387843175982733375, 13.25429856662670315938774607989, 14.356378622711175153785034085202, 15.39571954484944484822939917942, 16.30349694829974112399012905400, 17.80077312048599666735229714059, 18.711279636459443899071731486205, 20.20138436907227182925347438849, 21.164443028046816871514749966767, 22.16231528941964038001468326521, 22.73618732590257298202993770701, 24.72408267276093678291664608727, 25.564965234161918392303033781443, 26.32235103870330523979051648966, 27.54688883056552055648905478846, 28.4724671947752950264511363015, 29.532722033140963358197887047282, 30.87908225465112027781009806266

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