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Properties

Label 2-160-40.19-c4-0-20
Degree 22
Conductor 160160
Sign −0.780−0.624i-0.780 - 0.624i
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  − 10.2i·3-s + (17.9 − 17.4i)5-s − 84.8·7-s − 23.1·9-s − 71.3·11-s − 109.·13-s + (−177. − 182. i)15-s + 151. i·17-s + 368.·19-s + 865. i·21-s − 358.·23-s + (16.8 − 624. i)25-s − 590. i·27-s + 387. i·29-s + 1.68e3i·31-s + ⋯
L(s)  = 1  − 1.13i·3-s + (0.716 − 0.697i)5-s − 1.73·7-s − 0.285·9-s − 0.589·11-s − 0.646·13-s + (−0.790 − 0.812i)15-s + 0.523i·17-s + 1.02·19-s + 1.96i·21-s − 0.676·23-s + (0.0269 − 0.999i)25-s − 0.810i·27-s + 0.460i·29-s + 1.75i·31-s + ⋯

Functional equation

Λ(s)=(160s/2 ΓC(s) L(s)=((−0.780−0.624i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(160s/2 ΓC(s+2) L(s)=((−0.780−0.624i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: −0.780−0.624i-0.780 - 0.624i
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), −0.780−0.624i)(2,\ 160,\ (\ :2),\ -0.780 - 0.624i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.167450+0.477457i0.167450 + 0.477457i
L(12)L(\frac12) ≈\approx 0.167450+0.477457i0.167450 + 0.477457i
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+(−17.9+17.4i)T 1 + (-17.9 + 17.4i)T
good3 1+10.2iT−81T2 1 + 10.2iT - 81T^{2}
7 1+84.8T+2.40e3T2 1 + 84.8T + 2.40e3T^{2}
11 1+71.3T+1.46e4T2 1 + 71.3T + 1.46e4T^{2}
13 1+109.T+2.85e4T2 1 + 109.T + 2.85e4T^{2}
17 1−151.iT−8.35e4T2 1 - 151. iT - 8.35e4T^{2}
19 1−368.T+1.30e5T2 1 - 368.T + 1.30e5T^{2}
23 1+358.T+2.79e5T2 1 + 358.T + 2.79e5T^{2}
29 1−387.iT−7.07e5T2 1 - 387. iT - 7.07e5T^{2}
31 1−1.68e3iT−9.23e5T2 1 - 1.68e3iT - 9.23e5T^{2}
37 1+1.15e3T+1.87e6T2 1 + 1.15e3T + 1.87e6T^{2}
41 1+2.54e3T+2.82e6T2 1 + 2.54e3T + 2.82e6T^{2}
43 1+1.35e3iT−3.41e6T2 1 + 1.35e3iT - 3.41e6T^{2}
47 1+901.T+4.87e6T2 1 + 901.T + 4.87e6T^{2}
53 1+3.40e3T+7.89e6T2 1 + 3.40e3T + 7.89e6T^{2}
59 1−1.45e3T+1.21e7T2 1 - 1.45e3T + 1.21e7T^{2}
61 1+5.32e3iT−1.38e7T2 1 + 5.32e3iT - 1.38e7T^{2}
67 1−657.iT−2.01e7T2 1 - 657. iT - 2.01e7T^{2}
71 1+6.74e3iT−2.54e7T2 1 + 6.74e3iT - 2.54e7T^{2}
73 1−4.13e3iT−2.83e7T2 1 - 4.13e3iT - 2.83e7T^{2}
79 1+2.30e3iT−3.89e7T2 1 + 2.30e3iT - 3.89e7T^{2}
83 1−2.14e3iT−4.74e7T2 1 - 2.14e3iT - 4.74e7T^{2}
89 1−5.11e3T+6.27e7T2 1 - 5.11e3T + 6.27e7T^{2}
97 1+9.16e3iT−8.85e7T2 1 + 9.16e3iT - 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

−12.28392788282711924923646457859, −10.28678013216100631515210290454, −9.633611130069341825072229875556, −8.436278327604428446927356029757, −7.13385187525417763573819307870, −6.35784458249013753850175184130, −5.20719071277198823039279497246, −3.18681882498454037845828943716, −1.72602965161872832352622484820, −0.18151695912672010656135035554, 2.66883107020314396175123663635, 3.64215788105555989789201969431, 5.21202061316519386308961140350, 6.31221508928891963700396825934, 7.42355547421159676252372240396, 9.326848207539226517462108199509, 9.863249492008954125013468974274, 10.32094542274548057133967279960, 11.67087755936826612467748901875, 12.99587414395181417205107410465

Graph of the ZZ-function along the critical line