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Properties

Label 2-2e7-8.5-c7-0-5
Degree 22
Conductor 128128
Sign 11
Analytic cond. 39.985239.9852
Root an. cond. 6.323396.32339
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 82.2i·3-s + 232. i·5-s − 1.60e3·7-s − 4.57e3·9-s − 3.37e3i·11-s + 1.09e4i·13-s + 1.91e4·15-s + 2.31e4·17-s − 2.47e4i·19-s + 1.32e5i·21-s − 4.38e4·23-s + 2.40e4·25-s + 1.96e5i·27-s + 7.97e4i·29-s + 1.08e5·31-s + ⋯
L(s)  = 1  − 1.75i·3-s + 0.831i·5-s − 1.77·7-s − 2.09·9-s − 0.763i·11-s + 1.37i·13-s + 1.46·15-s + 1.14·17-s − 0.827i·19-s + 3.11i·21-s − 0.751·23-s + 0.307·25-s + 1.91i·27-s + 0.607i·29-s + 0.654·31-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=( Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+7/2) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 11
Analytic conductor: 39.985239.9852
Root analytic conductor: 6.323396.32339
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ128(65,⋅)\chi_{128} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :7/2), 1)(2,\ 128,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 1.0490254631.049025463
L(12)L(\frac12) ≈\approx 1.0490254631.049025463
L(92)L(\frac{9}{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 1
good3 1+82.2iT−2.18e3T2 1 + 82.2iT - 2.18e3T^{2}
5 1−232.iT−7.81e4T2 1 - 232. iT - 7.81e4T^{2}
7 1+1.60e3T+8.23e5T2 1 + 1.60e3T + 8.23e5T^{2}
11 1+3.37e3iT−1.94e7T2 1 + 3.37e3iT - 1.94e7T^{2}
13 1−1.09e4iT−6.27e7T2 1 - 1.09e4iT - 6.27e7T^{2}
17 1−2.31e4T+4.10e8T2 1 - 2.31e4T + 4.10e8T^{2}
19 1+2.47e4iT−8.93e8T2 1 + 2.47e4iT - 8.93e8T^{2}
23 1+4.38e4T+3.40e9T2 1 + 4.38e4T + 3.40e9T^{2}
29 1−7.97e4iT−1.72e10T2 1 - 7.97e4iT - 1.72e10T^{2}
31 1−1.08e5T+2.75e10T2 1 - 1.08e5T + 2.75e10T^{2}
37 1+1.73e5iT−9.49e10T2 1 + 1.73e5iT - 9.49e10T^{2}
41 1−4.34e4T+1.94e11T2 1 - 4.34e4T + 1.94e11T^{2}
43 1−6.09e5iT−2.71e11T2 1 - 6.09e5iT - 2.71e11T^{2}
47 1+3.18e4T+5.06e11T2 1 + 3.18e4T + 5.06e11T^{2}
53 1+1.98e6iT−1.17e12T2 1 + 1.98e6iT - 1.17e12T^{2}
59 1−1.92e6iT−2.48e12T2 1 - 1.92e6iT - 2.48e12T^{2}
61 1−1.63e6iT−3.14e12T2 1 - 1.63e6iT - 3.14e12T^{2}
67 1−1.97e6iT−6.06e12T2 1 - 1.97e6iT - 6.06e12T^{2}
71 1−3.01e6T+9.09e12T2 1 - 3.01e6T + 9.09e12T^{2}
73 1−2.39e6T+1.10e13T2 1 - 2.39e6T + 1.10e13T^{2}
79 1+2.37e6T+1.92e13T2 1 + 2.37e6T + 1.92e13T^{2}
83 1−2.95e6iT−2.71e13T2 1 - 2.95e6iT - 2.71e13T^{2}
89 1+7.18e6T+4.42e13T2 1 + 7.18e6T + 4.42e13T^{2}
97 1−1.49e7T+8.07e13T2 1 - 1.49e7T + 8.07e13T^{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

−12.16011015430612815887390014908, −11.26372255971652606153479209729, −9.852284619023557890555330231408, −8.658979442724825750217659810549, −7.26643069892315095393603121875, −6.64448266740675839889892121655, −5.96905331021797583736301986532, −3.37463821899618841132037612720, −2.46470447679173503291400570975, −0.863353193492562278944912926126, 0.40039045385511207751409877890, 3.02785745509486201113163611470, 3.88787100680286940941639866418, 5.14559617063660812147234781342, 6.06728520208443802084429180561, 8.005102284992406800525789284626, 9.236692046357756271924028647319, 9.989782798576779286203672621631, 10.35959133229660330484651330120, 12.14106588740784485504873790842

Graph of the ZZ-function along the critical line