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Properties

Label 2-2e3-8.5-c9-0-0
Degree 22
Conductor 88
Sign −0.957+0.287i-0.957 + 0.287i
Analytic cond. 4.120284.12028
Root an. cond. 2.029842.02984
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.36 + 20.6i)2-s + 150. i·3-s + (−336. − 385. i)4-s − 292. i·5-s + (−3.09e3 − 1.40e3i)6-s − 9.95e3·7-s + (1.10e4 − 3.32e3i)8-s − 2.84e3·9-s + (6.02e3 + 2.73e3i)10-s + 6.58e4i·11-s + (5.78e4 − 5.05e4i)12-s + 4.49e4i·13-s + (9.31e4 − 2.05e5i)14-s + 4.38e4·15-s + (−3.53e4 + 2.59e5i)16-s − 4.69e5·17-s + ⋯
L(s)  = 1  + (−0.413 + 0.910i)2-s + 1.06i·3-s + (−0.657 − 0.753i)4-s − 0.209i·5-s + (−0.974 − 0.442i)6-s − 1.56·7-s + (0.957 − 0.287i)8-s − 0.144·9-s + (0.190 + 0.0865i)10-s + 1.35i·11-s + (0.805 − 0.703i)12-s + 0.436i·13-s + (0.648 − 1.42i)14-s + 0.223·15-s + (−0.134 + 0.990i)16-s − 1.36·17-s + ⋯

Functional equation

Λ(s)=(8s/2 ΓC(s) L(s)=((−0.957+0.287i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(8s/2 ΓC(s+9/2) L(s)=((−0.957+0.287i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 88    =    232^{3}
Sign: −0.957+0.287i-0.957 + 0.287i
Analytic conductor: 4.120284.12028
Root analytic conductor: 2.029842.02984
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ8(5,⋅)\chi_{8} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 8, ( :9/2), −0.957+0.287i)(2,\ 8,\ (\ :9/2),\ -0.957 + 0.287i)

Particular Values

L(5)L(5) ≈\approx 0.0985981−0.672161i0.0985981 - 0.672161i
L(12)L(\frac12) ≈\approx 0.0985981−0.672161i0.0985981 - 0.672161i
L(112)L(\frac{11}{2}) 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+(9.36−20.6i)T 1 + (9.36 - 20.6i)T
good3 1−150.iT−1.96e4T2 1 - 150. iT - 1.96e4T^{2}
5 1+292.iT−1.95e6T2 1 + 292. iT - 1.95e6T^{2}
7 1+9.95e3T+4.03e7T2 1 + 9.95e3T + 4.03e7T^{2}
11 1−6.58e4iT−2.35e9T2 1 - 6.58e4iT - 2.35e9T^{2}
13 1−4.49e4iT−1.06e10T2 1 - 4.49e4iT - 1.06e10T^{2}
17 1+4.69e5T+1.18e11T2 1 + 4.69e5T + 1.18e11T^{2}
19 1−4.38e5iT−3.22e11T2 1 - 4.38e5iT - 3.22e11T^{2}
23 1−1.14e6T+1.80e12T2 1 - 1.14e6T + 1.80e12T^{2}
29 1+5.39e6iT−1.45e13T2 1 + 5.39e6iT - 1.45e13T^{2}
31 1−1.85e6T+2.64e13T2 1 - 1.85e6T + 2.64e13T^{2}
37 1−1.45e7iT−1.29e14T2 1 - 1.45e7iT - 1.29e14T^{2}
41 1−5.45e6T+3.27e14T2 1 - 5.45e6T + 3.27e14T^{2}
43 1+5.79e6iT−5.02e14T2 1 + 5.79e6iT - 5.02e14T^{2}
47 1+1.69e7T+1.11e15T2 1 + 1.69e7T + 1.11e15T^{2}
53 1−4.94e7iT−3.29e15T2 1 - 4.94e7iT - 3.29e15T^{2}
59 1+4.70e7iT−8.66e15T2 1 + 4.70e7iT - 8.66e15T^{2}
61 1+7.39e7iT−1.16e16T2 1 + 7.39e7iT - 1.16e16T^{2}
67 1−2.37e8iT−2.72e16T2 1 - 2.37e8iT - 2.72e16T^{2}
71 1+6.33e7T+4.58e16T2 1 + 6.33e7T + 4.58e16T^{2}
73 1+2.73e7T+5.88e16T2 1 + 2.73e7T + 5.88e16T^{2}
79 1+1.20e8T+1.19e17T2 1 + 1.20e8T + 1.19e17T^{2}
83 1+1.31e8iT−1.86e17T2 1 + 1.31e8iT - 1.86e17T^{2}
89 1−6.90e8T+3.50e17T2 1 - 6.90e8T + 3.50e17T^{2}
97 1−1.17e8T+7.60e17T2 1 - 1.17e8T + 7.60e17T^{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

−20.35654997087681474831710355049, −18.92058135327997373768219223363, −17.10262548445554492417815430900, −15.96687076876154894450233738908, −15.11543906891193679332604392158, −13.05816374540489534758214111123, −10.14137748426947771812555994469, −9.242654175021908540429993295571, −6.71858881119435499044570943558, −4.45881051517567841731270577497, 0.52850205214568098847237753075, 2.96656224618974369709787245977, 6.82589886433999997973353112969, 8.938761873834235855602559250976, 10.88419262385169586855391988775, 12.73574064298309083274962768521, 13.43118495372681160464466741231, 16.25782636158290861170907587868, 17.96764954421477419307056707701, 19.08551141617500367184553189494

Graph of the ZZ-function along the critical line