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Properties

Label 2-160-8.3-c4-0-13
Degree 22
Conductor 160160
Sign 0.563+0.825i0.563 + 0.825i
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  + 15.2·3-s − 11.1i·5-s − 47.5i·7-s + 150.·9-s − 84.0·11-s − 236. i·13-s − 170. i·15-s + 407.·17-s − 124.·19-s − 724. i·21-s − 198. i·23-s − 125.·25-s + 1.05e3·27-s + 1.31e3i·29-s + 1.24e3i·31-s + ⋯
L(s)  = 1  + 1.69·3-s − 0.447i·5-s − 0.971i·7-s + 1.85·9-s − 0.694·11-s − 1.39i·13-s − 0.756i·15-s + 1.41·17-s − 0.343·19-s − 1.64i·21-s − 0.375i·23-s − 0.200·25-s + 1.45·27-s + 1.56i·29-s + 1.29i·31-s + ⋯

Functional equation

Λ(s)=(160s/2 ΓC(s) L(s)=((0.563+0.825i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(160s/2 ΓC(s+2) L(s)=((0.563+0.825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: 0.563+0.825i0.563 + 0.825i
Analytic conductor: 16.539116.5391
Root analytic conductor: 4.066844.06684
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ160(111,⋅)\chi_{160} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 160, ( :2), 0.563+0.825i)(2,\ 160,\ (\ :2),\ 0.563 + 0.825i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.81979−1.48887i2.81979 - 1.48887i
L(12)L(\frac12) ≈\approx 2.81979−1.48887i2.81979 - 1.48887i
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+11.1iT 1 + 11.1iT
good3 1−15.2T+81T2 1 - 15.2T + 81T^{2}
7 1+47.5iT−2.40e3T2 1 + 47.5iT - 2.40e3T^{2}
11 1+84.0T+1.46e4T2 1 + 84.0T + 1.46e4T^{2}
13 1+236.iT−2.85e4T2 1 + 236. iT - 2.85e4T^{2}
17 1−407.T+8.35e4T2 1 - 407.T + 8.35e4T^{2}
19 1+124.T+1.30e5T2 1 + 124.T + 1.30e5T^{2}
23 1+198.iT−2.79e5T2 1 + 198. iT - 2.79e5T^{2}
29 1−1.31e3iT−7.07e5T2 1 - 1.31e3iT - 7.07e5T^{2}
31 1−1.24e3iT−9.23e5T2 1 - 1.24e3iT - 9.23e5T^{2}
37 1+854.iT−1.87e6T2 1 + 854. iT - 1.87e6T^{2}
41 1−3.07e3T+2.82e6T2 1 - 3.07e3T + 2.82e6T^{2}
43 1+1.10e3T+3.41e6T2 1 + 1.10e3T + 3.41e6T^{2}
47 1+284.iT−4.87e6T2 1 + 284. iT - 4.87e6T^{2}
53 1+2.96e3iT−7.89e6T2 1 + 2.96e3iT - 7.89e6T^{2}
59 1+657.T+1.21e7T2 1 + 657.T + 1.21e7T^{2}
61 1−4.20e3iT−1.38e7T2 1 - 4.20e3iT - 1.38e7T^{2}
67 1−2.60e3T+2.01e7T2 1 - 2.60e3T + 2.01e7T^{2}
71 1−4.54e3iT−2.54e7T2 1 - 4.54e3iT - 2.54e7T^{2}
73 1−822.T+2.83e7T2 1 - 822.T + 2.83e7T^{2}
79 1−3.04e3iT−3.89e7T2 1 - 3.04e3iT - 3.89e7T^{2}
83 1+1.12e4T+4.74e7T2 1 + 1.12e4T + 4.74e7T^{2}
89 1+4.38e3T+6.27e7T2 1 + 4.38e3T + 6.27e7T^{2}
97 1−4.44e3T+8.85e7T2 1 - 4.44e3T + 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

−12.61970898050435612076535118353, −10.60486923607831566451876028270, −10.00313146189513085215010428738, −8.764307666583623614565002744421, −7.961256428845415222593018065978, −7.25752943456632773854853158579, −5.26543386854645650030328396396, −3.78954566675677205474340707149, −2.83191625875302854734587890185, −1.09527278477892981191534660709, 2.01560084676799809241617362572, 2.90005481620528864281528410772, 4.20270093397125306724210551827, 5.97487581217592451019462935841, 7.47702392880393656789363282611, 8.208276950879684494079987872539, 9.301019596331628750364258368419, 9.888999833567989635309483439165, 11.43268054214140075788009219411, 12.51988787064466894327864516092

Graph of the ZZ-function along the critical line