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Properties

Label 2-160-8.5-c5-0-13
Degree 22
Conductor 160160
Sign −0.0379+0.999i-0.0379 + 0.999i
Analytic cond. 25.661425.6614
Root an. cond. 5.065705.06570
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 18.7i·3-s + 25i·5-s + 107.·7-s − 109.·9-s − 272. i·11-s − 198. i·13-s + 469.·15-s + 2.06e3·17-s + 1.89e3i·19-s − 2.02e3i·21-s + 987.·23-s − 625·25-s − 2.49e3i·27-s − 8.01e3i·29-s − 827.·31-s + ⋯
L(s)  = 1  − 1.20i·3-s + 0.447i·5-s + 0.829·7-s − 0.452·9-s − 0.678i·11-s − 0.325i·13-s + 0.538·15-s + 1.73·17-s + 1.20i·19-s − 0.999i·21-s + 0.389·23-s − 0.200·25-s − 0.659i·27-s − 1.76i·29-s − 0.154·31-s + ⋯

Functional equation

Λ(s)=(160s/2 ΓC(s) L(s)=((−0.0379+0.999i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0379 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(160s/2 ΓC(s+5/2) L(s)=((−0.0379+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0379 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: −0.0379+0.999i-0.0379 + 0.999i
Analytic conductor: 25.661425.6614
Root analytic conductor: 5.065705.06570
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ160(81,⋅)\chi_{160} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 160, ( :5/2), −0.0379+0.999i)(2,\ 160,\ (\ :5/2),\ -0.0379 + 0.999i)

Particular Values

L(3)L(3) ≈\approx 2.1685487892.168548789
L(12)L(\frac12) ≈\approx 2.1685487892.168548789
L(72)L(\frac{7}{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
5 1−25iT 1 - 25iT
good3 1+18.7iT−243T2 1 + 18.7iT - 243T^{2}
7 1−107.T+1.68e4T2 1 - 107.T + 1.68e4T^{2}
11 1+272.iT−1.61e5T2 1 + 272. iT - 1.61e5T^{2}
13 1+198.iT−3.71e5T2 1 + 198. iT - 3.71e5T^{2}
17 1−2.06e3T+1.41e6T2 1 - 2.06e3T + 1.41e6T^{2}
19 1−1.89e3iT−2.47e6T2 1 - 1.89e3iT - 2.47e6T^{2}
23 1−987.T+6.43e6T2 1 - 987.T + 6.43e6T^{2}
29 1+8.01e3iT−2.05e7T2 1 + 8.01e3iT - 2.05e7T^{2}
31 1+827.T+2.86e7T2 1 + 827.T + 2.86e7T^{2}
37 1+9.42e3iT−6.93e7T2 1 + 9.42e3iT - 6.93e7T^{2}
41 1+8.22e3T+1.15e8T2 1 + 8.22e3T + 1.15e8T^{2}
43 1+9.30e3iT−1.47e8T2 1 + 9.30e3iT - 1.47e8T^{2}
47 1+1.38e4T+2.29e8T2 1 + 1.38e4T + 2.29e8T^{2}
53 1−2.77e4iT−4.18e8T2 1 - 2.77e4iT - 4.18e8T^{2}
59 1+2.51e4iT−7.14e8T2 1 + 2.51e4iT - 7.14e8T^{2}
61 1+2.64e4iT−8.44e8T2 1 + 2.64e4iT - 8.44e8T^{2}
67 1+3.85e4iT−1.35e9T2 1 + 3.85e4iT - 1.35e9T^{2}
71 1−7.10e4T+1.80e9T2 1 - 7.10e4T + 1.80e9T^{2}
73 1−1.86e4T+2.07e9T2 1 - 1.86e4T + 2.07e9T^{2}
79 1−7.55e4T+3.07e9T2 1 - 7.55e4T + 3.07e9T^{2}
83 1−1.25e5iT−3.93e9T2 1 - 1.25e5iT - 3.93e9T^{2}
89 1−3.03e4T+5.58e9T2 1 - 3.03e4T + 5.58e9T^{2}
97 1−1.56e4T+8.58e9T2 1 - 1.56e4T + 8.58e9T^{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

−11.90994968198313637948918134300, −10.89421894015479374536041833194, −9.770646272679073228854407939107, −8.008172441713448161038963641878, −7.81814537580731995966336514959, −6.42280467502948310775648461789, −5.43359722920422425353354023690, −3.55815803645921964863397319429, −2.00184520601613364558815717172, −0.812333429012405914020304403537, 1.38772710423946496734483801737, 3.33409452741340317732084688745, 4.71841184765186675352112704200, 5.18180160510757385868484675776, 7.04544023614676914730362864852, 8.308278003008287067442571447065, 9.345076426365460052177829730855, 10.12873203381046433562453432118, 11.12812282800311555164704427073, 12.08422184116632829463425231928

Graph of the ZZ-function along the critical line