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Properties

Label 2-162-3.2-c6-0-20
Degree 22
Conductor 162162
Sign −1-1
Analytic cond. 37.268737.2687
Root an. cond. 6.104816.10481
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65i·2-s − 32.0·4-s − 110. i·5-s + 326.·7-s + 181. i·8-s − 626.·10-s − 625. i·11-s − 2.02e3·13-s − 1.84e3i·14-s + 1.02e3·16-s − 4.12e3i·17-s + 1.21e4·19-s + 3.54e3i·20-s − 3.53e3·22-s − 2.14e4i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s − 0.885i·5-s + 0.951·7-s + 0.353i·8-s − 0.626·10-s − 0.470i·11-s − 0.923·13-s − 0.672i·14-s + 0.250·16-s − 0.839i·17-s + 1.77·19-s + 0.442i·20-s − 0.332·22-s − 1.76i·23-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=(− Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+3) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: −1-1
Analytic conductor: 37.268737.2687
Root analytic conductor: 6.104816.10481
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ162(161,⋅)\chi_{162} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :3), −1)(2,\ 162,\ (\ :3),\ -1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.4729069661.472906966
L(12)L(\frac12) ≈\approx 1.4729069661.472906966
L(4)L(4) 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+5.65iT 1 + 5.65iT
3 1 1
good5 1+110.iT−1.56e4T2 1 + 110. iT - 1.56e4T^{2}
7 1−326.T+1.17e5T2 1 - 326.T + 1.17e5T^{2}
11 1+625.iT−1.77e6T2 1 + 625. iT - 1.77e6T^{2}
13 1+2.02e3T+4.82e6T2 1 + 2.02e3T + 4.82e6T^{2}
17 1+4.12e3iT−2.41e7T2 1 + 4.12e3iT - 2.41e7T^{2}
19 1−1.21e4T+4.70e7T2 1 - 1.21e4T + 4.70e7T^{2}
23 1+2.14e4iT−1.48e8T2 1 + 2.14e4iT - 1.48e8T^{2}
29 1−2.47e4iT−5.94e8T2 1 - 2.47e4iT - 5.94e8T^{2}
31 1+4.05e4T+8.87e8T2 1 + 4.05e4T + 8.87e8T^{2}
37 1+3.00e4T+2.56e9T2 1 + 3.00e4T + 2.56e9T^{2}
41 1−5.92e4iT−4.75e9T2 1 - 5.92e4iT - 4.75e9T^{2}
43 1+8.79e4T+6.32e9T2 1 + 8.79e4T + 6.32e9T^{2}
47 1+1.29e5iT−1.07e10T2 1 + 1.29e5iT - 1.07e10T^{2}
53 1+1.65e5iT−2.21e10T2 1 + 1.65e5iT - 2.21e10T^{2}
59 1−5.32e4iT−4.21e10T2 1 - 5.32e4iT - 4.21e10T^{2}
61 1+2.66e5T+5.15e10T2 1 + 2.66e5T + 5.15e10T^{2}
67 1+4.07e5T+9.04e10T2 1 + 4.07e5T + 9.04e10T^{2}
71 1−1.86e5iT−1.28e11T2 1 - 1.86e5iT - 1.28e11T^{2}
73 1+2.42e5T+1.51e11T2 1 + 2.42e5T + 1.51e11T^{2}
79 1+1.26e5T+2.43e11T2 1 + 1.26e5T + 2.43e11T^{2}
83 1−6.89e4iT−3.26e11T2 1 - 6.89e4iT - 3.26e11T^{2}
89 1−4.13e5iT−4.96e11T2 1 - 4.13e5iT - 4.96e11T^{2}
97 1−9.78e5T+8.32e11T2 1 - 9.78e5T + 8.32e11T^{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

−11.43931097440848842179517262966, −10.30383826747226196557663939531, −9.196081630002515322763283421803, −8.398997152021743380889055494441, −7.22634204834298859785296830750, −5.24901292799266953203470567938, −4.75386448346962458356244972214, −3.08819265673859249241345467081, −1.59575459026104394912794821570, −0.43685981030268812081377515006, 1.64968759588892319160617253438, 3.33990678986339634890996608692, 4.81934590504708385098520478609, 5.83674871636874586217008859408, 7.35464763393777658979514575014, 7.62735640825490909454941090713, 9.177044171455404847372426014928, 10.14043141109318585100238098880, 11.25920613367504847194209501960, 12.17951607101778170540382843210

Graph of the ZZ-function along the critical line