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Properties

Label 2-160-40.19-c4-0-3
Degree 22
Conductor 160160
Sign 0.256−0.966i0.256 - 0.966i
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  − 10.5i·3-s + (−24.9 + 1.70i)5-s − 26.0·7-s − 30.0·9-s + 94.0·11-s − 236.·13-s + (18.0 + 262. i)15-s + 398. i·17-s − 98.0·19-s + 274. i·21-s + 490.·23-s + (619. − 85.2i)25-s − 537. i·27-s + 1.45e3i·29-s + 442. i·31-s + ⋯
L(s)  = 1  − 1.17i·3-s + (−0.997 + 0.0683i)5-s − 0.530·7-s − 0.370·9-s + 0.776·11-s − 1.39·13-s + (0.0800 + 1.16i)15-s + 1.37i·17-s − 0.271·19-s + 0.621i·21-s + 0.926·23-s + (0.990 − 0.136i)25-s − 0.736i·27-s + 1.72i·29-s + 0.460i·31-s + ⋯

Functional equation

Λ(s)=(160s/2 ΓC(s) L(s)=((0.256−0.966i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(160s/2 ΓC(s+2) L(s)=((0.256−0.966i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: 0.256−0.966i0.256 - 0.966i
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), 0.256−0.966i)(2,\ 160,\ (\ :2),\ 0.256 - 0.966i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.463039+0.356115i0.463039 + 0.356115i
L(12)L(\frac12) ≈\approx 0.463039+0.356115i0.463039 + 0.356115i
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+(24.9−1.70i)T 1 + (24.9 - 1.70i)T
good3 1+10.5iT−81T2 1 + 10.5iT - 81T^{2}
7 1+26.0T+2.40e3T2 1 + 26.0T + 2.40e3T^{2}
11 1−94.0T+1.46e4T2 1 - 94.0T + 1.46e4T^{2}
13 1+236.T+2.85e4T2 1 + 236.T + 2.85e4T^{2}
17 1−398.iT−8.35e4T2 1 - 398. iT - 8.35e4T^{2}
19 1+98.0T+1.30e5T2 1 + 98.0T + 1.30e5T^{2}
23 1−490.T+2.79e5T2 1 - 490.T + 2.79e5T^{2}
29 1−1.45e3iT−7.07e5T2 1 - 1.45e3iT - 7.07e5T^{2}
31 1−442.iT−9.23e5T2 1 - 442. iT - 9.23e5T^{2}
37 1−1.21e3T+1.87e6T2 1 - 1.21e3T + 1.87e6T^{2}
41 1−336.T+2.82e6T2 1 - 336.T + 2.82e6T^{2}
43 1−1.73e3iT−3.41e6T2 1 - 1.73e3iT - 3.41e6T^{2}
47 1+1.94e3T+4.87e6T2 1 + 1.94e3T + 4.87e6T^{2}
53 1+2.00e3T+7.89e6T2 1 + 2.00e3T + 7.89e6T^{2}
59 1+6.52e3T+1.21e7T2 1 + 6.52e3T + 1.21e7T^{2}
61 1−4.99e3iT−1.38e7T2 1 - 4.99e3iT - 1.38e7T^{2}
67 1−5.38e3iT−2.01e7T2 1 - 5.38e3iT - 2.01e7T^{2}
71 1+6.60e3iT−2.54e7T2 1 + 6.60e3iT - 2.54e7T^{2}
73 1+270.iT−2.83e7T2 1 + 270. iT - 2.83e7T^{2}
79 1+1.07e3iT−3.89e7T2 1 + 1.07e3iT - 3.89e7T^{2}
83 1+8.97e3iT−4.74e7T2 1 + 8.97e3iT - 4.74e7T^{2}
89 1+3.55e3T+6.27e7T2 1 + 3.55e3T + 6.27e7T^{2}
97 1−8.43e3iT−8.85e7T2 1 - 8.43e3iT - 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

−12.59671100483607205454716357022, −11.70761947148504294037014586729, −10.55100761564942250848321394748, −9.174805693537658727160759043705, −8.005303677852058131404659660228, −7.12986057950396900208154328190, −6.38358413737659119532289845706, −4.56812972259706020653449365419, −3.09874432652488663315335688618, −1.37304727610956679034331558212, 0.23840793402052761850912064816, 2.96228146386821394436950936226, 4.18846137726065917426131484508, 4.97380094307882125875982270700, 6.75152598901803664652428676622, 7.81062601656837079462888831173, 9.338427991668426263347252931148, 9.697024922329466916211159748058, 11.04099737893628821142468935835, 11.82104178106900614628281461436

Graph of the ZZ-function along the critical line