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Properties

Label 2-160-8.3-c4-0-6
Degree 22
Conductor 160160
Sign 0.457+0.889i0.457 + 0.889i
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  − 13.1·3-s − 11.1i·5-s + 61.6i·7-s + 91.5·9-s − 182.·11-s + 141. i·13-s + 146. i·15-s + 40.3·17-s + 321.·19-s − 809. i·21-s − 816. i·23-s − 125.·25-s − 138.·27-s − 1.19e3i·29-s − 361. i·31-s + ⋯
L(s)  = 1  − 1.45·3-s − 0.447i·5-s + 1.25i·7-s + 1.12·9-s − 1.50·11-s + 0.838i·13-s + 0.652i·15-s + 0.139·17-s + 0.890·19-s − 1.83i·21-s − 1.54i·23-s − 0.200·25-s − 0.189·27-s − 1.42i·29-s − 0.376i·31-s + ⋯

Functional equation

Λ(s)=(160s/2 ΓC(s) L(s)=((0.457+0.889i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(160s/2 ΓC(s+2) L(s)=((0.457+0.889i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 160160    =    25⋅52^{5} \cdot 5
Sign: 0.457+0.889i0.457 + 0.889i
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), 0.457+0.889i)(2,\ 160,\ (\ :2),\ 0.457 + 0.889i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.541155−0.330128i0.541155 - 0.330128i
L(12)L(\frac12) ≈\approx 0.541155−0.330128i0.541155 - 0.330128i
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+13.1T+81T2 1 + 13.1T + 81T^{2}
7 1−61.6iT−2.40e3T2 1 - 61.6iT - 2.40e3T^{2}
11 1+182.T+1.46e4T2 1 + 182.T + 1.46e4T^{2}
13 1−141.iT−2.85e4T2 1 - 141. iT - 2.85e4T^{2}
17 1−40.3T+8.35e4T2 1 - 40.3T + 8.35e4T^{2}
19 1−321.T+1.30e5T2 1 - 321.T + 1.30e5T^{2}
23 1+816.iT−2.79e5T2 1 + 816. iT - 2.79e5T^{2}
29 1+1.19e3iT−7.07e5T2 1 + 1.19e3iT - 7.07e5T^{2}
31 1+361.iT−9.23e5T2 1 + 361. iT - 9.23e5T^{2}
37 1−0.0209iT−1.87e6T2 1 - 0.0209iT - 1.87e6T^{2}
41 1+872.T+2.82e6T2 1 + 872.T + 2.82e6T^{2}
43 1−3.12e3T+3.41e6T2 1 - 3.12e3T + 3.41e6T^{2}
47 1+2.65e3iT−4.87e6T2 1 + 2.65e3iT - 4.87e6T^{2}
53 1−2.95e3iT−7.89e6T2 1 - 2.95e3iT - 7.89e6T^{2}
59 1+5.41e3T+1.21e7T2 1 + 5.41e3T + 1.21e7T^{2}
61 1+2.25e3iT−1.38e7T2 1 + 2.25e3iT - 1.38e7T^{2}
67 1−2.35e3T+2.01e7T2 1 - 2.35e3T + 2.01e7T^{2}
71 1−660.iT−2.54e7T2 1 - 660. iT - 2.54e7T^{2}
73 1−5.94e3T+2.83e7T2 1 - 5.94e3T + 2.83e7T^{2}
79 1+7.62e3iT−3.89e7T2 1 + 7.62e3iT - 3.89e7T^{2}
83 1−3.52e3T+4.74e7T2 1 - 3.52e3T + 4.74e7T^{2}
89 1−5.94e3T+6.27e7T2 1 - 5.94e3T + 6.27e7T^{2}
97 1−679.T+8.85e7T2 1 - 679.T + 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

−12.07672979004041737588129054602, −11.19843954172903644838279398603, −10.19225713283822550846498858367, −9.024600930255266165135208329798, −7.79693810143605969583491180185, −6.29157815842528301609014980867, −5.49611788690343499880098653431, −4.65811724952130210213876130767, −2.38927414778842235297033757381, −0.38998675022357131634049251470, 0.935077174325273511037742052353, 3.29672324930304273493700856051, 4.96318125525992865330498844009, 5.73666691958518896022383652299, 7.11655664347921083336644614386, 7.76919808293364890403250845857, 9.813348804580365998433956354305, 10.71372145770503831804412093343, 11.01711840154364694617738529203, 12.31675456902400426398017459960

Graph of the ZZ-function along the critical line