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Properties

Label 2-88-88.27-c0-0-0
Degree 22
Conductor 8888
Sign 0.624−0.781i0.624 - 0.781i
Analytic cond. 0.04391770.0439177
Root an. cond. 0.2095650.209565
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(88s/2 ΓC(s) L(s)=((0.624−0.781i) Λ‾(1−s)\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}
Λ(s)=(88s/2 ΓC(s) L(s)=((0.624−0.781i) Λ‾(1−s)\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: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: 0.624−0.781i0.624 - 0.781i
Analytic conductor: 0.04391770.0439177
Root analytic conductor: 0.2095650.209565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ88(27,⋅)\chi_{88} (27, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 88, ( :0), 0.624−0.781i)(2,\ 88,\ (\ :0),\ 0.624 - 0.781i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50206178440.5020617844
L(12)L(\frac12) ≈\approx 0.50206178440.5020617844
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
11 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
good3 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
5 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
7 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
13 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
17 1+(0.5−1.53i)T+(−0.809−0.587i)T2 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}
19 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
23 1−T2 1 - T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
31 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
37 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
41 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
43 1−0.618T+T2 1 - 0.618T + T^{2}
47 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
53 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
59 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}
61 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
67 1+1.61T+T2 1 + 1.61T + T^{2}
71 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
73 1+(0.5−0.363i)T+(0.309−0.951i)T2 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}
79 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
83 1+(−0.190+0.587i)T+(−0.809−0.587i)T2 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}
89 1−0.618T+T2 1 - 0.618T + T^{2}
97 1+(−0.190−0.587i)T+(−0.809+0.587i)T2 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line