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Properties

Label 2-88-88.27-c0-0-0
Degree $2$
Conductor $88$
Sign $0.624 - 0.781i$
Analytic cond. $0.0439177$
Root an. cond. $0.209565$
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)2-s + (−0.5 − 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.190 − 0.587i)9-s + (−0.809 − 0.587i)11-s + 0.618·12-s + (0.309 − 0.951i)16-s + (−0.5 + 1.53i)17-s + (0.5 − 0.363i)18-s + (1.30 + 0.951i)19-s + (0.309 − 0.951i)22-s + (0.190 + 0.587i)24-s + (−0.809 − 0.587i)25-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.5 − 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (−0.809 − 0.587i)8-s + (−0.190 − 0.587i)9-s + (−0.809 − 0.587i)11-s + 0.618·12-s + (0.309 − 0.951i)16-s + (−0.5 + 1.53i)17-s + (0.5 − 0.363i)18-s + (1.30 + 0.951i)19-s + (0.309 − 0.951i)22-s + (0.190 + 0.587i)24-s + (−0.809 − 0.587i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.624 - 0.781i$
Analytic conductor: \(0.0439177\)
Root analytic conductor: \(0.209565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :0),\ 0.624 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5020617844\)
\(L(\frac12)\) \(\approx\) \(0.5020617844\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.63897058595544622008336749038, −13.57082369504535396098154693607, −12.63529772308978450025900787089, −11.68522582067869304824667459207, −10.15619435925391413893199930499, −8.700097417800058409783457980085, −7.66367238784448783056406444752, −6.29990722230209404329316790795, −5.50038095571044670886224694877, −3.67884878259526706903463378333, 2.68791914585162550379183914391, 4.65735988994615014311933168328, 5.47635468920967000080472360694, 7.48952021262817008111292203391, 9.191973812477788515768121620020, 10.13125239066839070428183229427, 11.19988117789173054107974827317, 11.86431717417097928868419946550, 13.27116718676973756489086046972, 13.87047523104848998121753688517

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