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Properties

Label 1-88-88.5-r0-0-0
Degree $1$
Conductor $88$
Sign $0.550 - 0.835i$
Analytic cond. $0.408670$
Root an. cond. $0.408670$
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

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

Invariants

Degree: \(1\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.550 - 0.835i$
Analytic conductor: \(0.408670\)
Root analytic conductor: \(0.408670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 88,\ (0:\ ),\ 0.550 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8873123181 - 0.4779865549i\)
\(L(\frac12)\) \(\approx\) \(0.8873123181 - 0.4779865549i\)
\(L(1)\) \(\approx\) \(1.001420540 - 0.3227405149i\)
\(L(1)\) \(\approx\) \(1.001420540 - 0.3227405149i\)

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

−31.02652445292420804418057384800, −29.35106018367485817750632756029, −28.4412015651450240094808394495, −27.88456088410504525301877431852, −26.55974144906941114936055059899, −25.50802110289202316829722305691, −24.53724835795915504411979224066, −23.19569902821563278588781011388, −21.91927595150137256852535890384, −21.25060212890845185863158748416, −20.46162621346778344028661022340, −18.76545533252955094572267717068, −17.553312413568745776677832899461, −16.63595796036717431838305712184, −15.53938915139503183164600164581, −14.47927988695861736536003940929, −13.059426931199010889492380000220, −11.74516962601399400143715055862, −10.58675809399275808096180260214, −9.24729213184655978584207487491, −8.596617195202677711721682097473, −6.194815391562907522742974636726, −5.341810199486156004868378140194, −3.99746052763048929477050903683, −2.0402878516846803363646896500, 1.36391045074167694049536803234, 2.95769092282730251397806884138, 5.07885796113543017750302778475, 6.513973490625909545861746738274, 7.29144661096309426908815016015, 8.83869567507338897651220114347, 10.59233933498749200573158606595, 11.27250328462869604211085300400, 13.1123906325926113623702561752, 13.58339224555293266776845042573, 14.833893542102604704918239361, 16.60119965734599724555729547701, 17.703881316413473183376618770281, 18.222087230868698300362340962206, 19.60660805317336798803179186391, 20.68383291008391938717770623390, 22.11150354881355887057883427140, 23.0652868225934858926999143724, 24.004203274344361597612995481788, 25.169824596028928880488958956769, 25.98918555782680332776654256532, 27.243159204910453363453460425358, 28.64016833973288808684928223721, 29.52165659944778518926318249598, 30.25202639817529024507248100887

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