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Properties

Label 2-160-40.19-c4-0-17
Degree 22
Conductor 160160
Sign −0.345+0.938i-0.345 + 0.938i
Analytic cond. 16.539116.5391
Root an. cond. 4.066844.06684
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.05i·3-s + (−4.74 − 24.5i)5-s + 65.1·7-s − 0.965·9-s + 220.·11-s − 75.1·13-s + (−222. + 42.9i)15-s − 341. i·17-s + 59.8·19-s − 590. i·21-s − 449.·23-s + (−579. + 233. i)25-s − 724. i·27-s + 977. i·29-s − 89.3i·31-s + ⋯
L(s)  = 1  − 1.00i·3-s + (−0.189 − 0.981i)5-s + 1.33·7-s − 0.0119·9-s + 1.81·11-s − 0.444·13-s + (−0.987 + 0.191i)15-s − 1.18i·17-s + 0.165·19-s − 1.33i·21-s − 0.848·23-s + (−0.927 + 0.372i)25-s − 0.993i·27-s + 1.16i·29-s − 0.0930i·31-s + ⋯

Functional equation

Λ(s)=(160s/2 ΓC(s) L(s)=((−0.345+0.938i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(160s/2 ΓC(s+2) L(s)=((−0.345+0.938i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: −0.345+0.938i-0.345 + 0.938i
Analytic conductor: 16.539116.5391
Root analytic conductor: 4.066844.06684
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ160(79,⋅)\chi_{160} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 160, ( :2), −0.345+0.938i)(2,\ 160,\ (\ :2),\ -0.345 + 0.938i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.25303−1.79620i1.25303 - 1.79620i
L(12)L(\frac12) ≈\approx 1.25303−1.79620i1.25303 - 1.79620i
L(3)L(3) 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+(4.74+24.5i)T 1 + (4.74 + 24.5i)T
good3 1+9.05iT−81T2 1 + 9.05iT - 81T^{2}
7 1−65.1T+2.40e3T2 1 - 65.1T + 2.40e3T^{2}
11 1−220.T+1.46e4T2 1 - 220.T + 1.46e4T^{2}
13 1+75.1T+2.85e4T2 1 + 75.1T + 2.85e4T^{2}
17 1+341.iT−8.35e4T2 1 + 341. iT - 8.35e4T^{2}
19 1−59.8T+1.30e5T2 1 - 59.8T + 1.30e5T^{2}
23 1+449.T+2.79e5T2 1 + 449.T + 2.79e5T^{2}
29 1−977.iT−7.07e5T2 1 - 977. iT - 7.07e5T^{2}
31 1+89.3iT−9.23e5T2 1 + 89.3iT - 9.23e5T^{2}
37 1+1.48e3T+1.87e6T2 1 + 1.48e3T + 1.87e6T^{2}
41 1−1.55e3T+2.82e6T2 1 - 1.55e3T + 2.82e6T^{2}
43 1−245.iT−3.41e6T2 1 - 245. iT - 3.41e6T^{2}
47 1+180.T+4.87e6T2 1 + 180.T + 4.87e6T^{2}
53 1+1.24e3T+7.89e6T2 1 + 1.24e3T + 7.89e6T^{2}
59 1−4.05e3T+1.21e7T2 1 - 4.05e3T + 1.21e7T^{2}
61 1+3.52e3iT−1.38e7T2 1 + 3.52e3iT - 1.38e7T^{2}
67 1+3.15e3iT−2.01e7T2 1 + 3.15e3iT - 2.01e7T^{2}
71 1−5.61e3iT−2.54e7T2 1 - 5.61e3iT - 2.54e7T^{2}
73 1−2.40e3iT−2.83e7T2 1 - 2.40e3iT - 2.83e7T^{2}
79 1−3.92e3iT−3.89e7T2 1 - 3.92e3iT - 3.89e7T^{2}
83 1−7.72e3iT−4.74e7T2 1 - 7.72e3iT - 4.74e7T^{2}
89 1+4.58e3T+6.27e7T2 1 + 4.58e3T + 6.27e7T^{2}
97 1−2.35e3iT−8.85e7T2 1 - 2.35e3iT - 8.85e7T^{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

−11.96409141109020663083231532299, −11.38323464294206876182760576826, −9.611656793132337134041806913358, −8.646232718404661463420982128475, −7.68576354622029040844503071167, −6.75872261031382185871641627364, −5.22344183722650459729982530236, −4.14940778447766177556520834056, −1.81218388774105864751531978494, −0.959677945508520984684914031354, 1.77063943422811775685127513364, 3.74207696618920689716302825567, 4.44326426306121711087363683282, 6.02054833407971857167847747032, 7.27505506578439355350797623365, 8.456144482677829230119699963696, 9.647759915907638705395035805085, 10.51854921037599929382100348941, 11.38131623897420597271944748200, 12.11368985462837262840458749872

Graph of the ZZ-function along the critical line