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Properties

Label 2-99-11.10-c8-0-33
Degree 22
Conductor 9999
Sign −0.734+0.679i-0.734 + 0.679i
Analytic cond. 40.330440.3304
Root an. cond. 6.350626.35062
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.5i·2-s + 43.4·4-s + 870.·5-s − 4.07e3i·7-s − 4.36e3i·8-s − 1.26e4i·10-s + (1.07e4 − 9.94e3i)11-s + 3.71e4i·13-s − 5.94e4·14-s − 5.25e4·16-s − 2.00e3i·17-s − 8.95e4i·19-s + 3.78e4·20-s + (−1.44e5 − 1.56e5i)22-s + 2.77e5·23-s + ⋯
L(s)  = 1  − 0.911i·2-s + 0.169·4-s + 1.39·5-s − 1.69i·7-s − 1.06i·8-s − 1.26i·10-s + (0.734 − 0.679i)11-s + 1.29i·13-s − 1.54·14-s − 0.801·16-s − 0.0239i·17-s − 0.686i·19-s + 0.236·20-s + (−0.618 − 0.668i)22-s + 0.991·23-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=((−0.734+0.679i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+4) L(s)=((−0.734+0.679i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: −0.734+0.679i-0.734 + 0.679i
Analytic conductor: 40.330440.3304
Root analytic conductor: 6.350626.35062
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ99(10,⋅)\chi_{99} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 99, ( :4), −0.734+0.679i)(2,\ 99,\ (\ :4),\ -0.734 + 0.679i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.17000−2.98813i1.17000 - 2.98813i
L(12)L(\frac12) ≈\approx 1.17000−2.98813i1.17000 - 2.98813i
L(5)L(5) 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)
bad3 1 1
11 1+(−1.07e4+9.94e3i)T 1 + (-1.07e4 + 9.94e3i)T
good2 1+14.5iT−256T2 1 + 14.5iT - 256T^{2}
5 1−870.T+3.90e5T2 1 - 870.T + 3.90e5T^{2}
7 1+4.07e3iT−5.76e6T2 1 + 4.07e3iT - 5.76e6T^{2}
13 1−3.71e4iT−8.15e8T2 1 - 3.71e4iT - 8.15e8T^{2}
17 1+2.00e3iT−6.97e9T2 1 + 2.00e3iT - 6.97e9T^{2}
19 1+8.95e4iT−1.69e10T2 1 + 8.95e4iT - 1.69e10T^{2}
23 1−2.77e5T+7.83e10T2 1 - 2.77e5T + 7.83e10T^{2}
29 1−3.25e5iT−5.00e11T2 1 - 3.25e5iT - 5.00e11T^{2}
31 1−4.44e5T+8.52e11T2 1 - 4.44e5T + 8.52e11T^{2}
37 1+1.48e6T+3.51e12T2 1 + 1.48e6T + 3.51e12T^{2}
41 1−1.84e6iT−7.98e12T2 1 - 1.84e6iT - 7.98e12T^{2}
43 1−3.67e6iT−1.16e13T2 1 - 3.67e6iT - 1.16e13T^{2}
47 1+3.89e6T+2.38e13T2 1 + 3.89e6T + 2.38e13T^{2}
53 1−6.99e6T+6.22e13T2 1 - 6.99e6T + 6.22e13T^{2}
59 1−4.71e6T+1.46e14T2 1 - 4.71e6T + 1.46e14T^{2}
61 1+7.61e6iT−1.91e14T2 1 + 7.61e6iT - 1.91e14T^{2}
67 1+1.80e7T+4.06e14T2 1 + 1.80e7T + 4.06e14T^{2}
71 1+4.47e6T+6.45e14T2 1 + 4.47e6T + 6.45e14T^{2}
73 1−1.06e7iT−8.06e14T2 1 - 1.06e7iT - 8.06e14T^{2}
79 1+4.97e7iT−1.51e15T2 1 + 4.97e7iT - 1.51e15T^{2}
83 1+6.38e7iT−2.25e15T2 1 + 6.38e7iT - 2.25e15T^{2}
89 1−1.01e7T+3.93e15T2 1 - 1.01e7T + 3.93e15T^{2}
97 1+1.28e8T+7.83e15T2 1 + 1.28e8T + 7.83e15T^{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

−11.60092164781385075318535719148, −10.82351372757761911721038739892, −9.944993091021456841343648734977, −9.091886377470389108545928165433, −7.04159608087420884538579477258, −6.40321135869583151329936099388, −4.50102246356379774366585017404, −3.21478796037658261159979614970, −1.71502653057883163241973697329, −0.917956235821705968884873063862, 1.76080489384544735113173241391, 2.71050161091133845198043980041, 5.32114651511740927078072723390, 5.77935345543696829891306812546, 6.82273925341447463614359371851, 8.323401425096496482634023610927, 9.238099140431217054683070163791, 10.34209165488236186277411005704, 11.82399651328201115441416959545, 12.71732506944084370064933263246

Graph of the ZZ-function along the critical line