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Properties

Label 2-162-9.2-c6-0-6
Degree 22
Conductor 162162
Sign 0.996−0.0871i0.996 - 0.0871i
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  + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (−180. + 104. i)5-s + (−2.09 + 3.63i)7-s − 181. i·8-s + 1.17e3·10-s + (−1.95e3 − 1.13e3i)11-s + (−1.42e3 − 2.45e3i)13-s + (20.5 − 11.8i)14-s + (−512. + 886. i)16-s − 1.96e3i·17-s − 281.·19-s + (−5.77e3 − 3.33e3i)20-s + (6.39e3 + 1.10e4i)22-s + (−1.45e4 + 8.37e3i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.44 + 0.834i)5-s + (−0.00611 + 0.0105i)7-s − 0.353i·8-s + 1.17·10-s + (−1.47 − 0.849i)11-s + (−0.646 − 1.11i)13-s + (0.00749 − 0.00432i)14-s + (−0.125 + 0.216i)16-s − 0.400i·17-s − 0.0410·19-s + (−0.722 − 0.417i)20-s + (0.600 + 1.04i)22-s + (−1.19 + 0.688i)23-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((0.996−0.0871i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+3) L(s)=((0.996−0.0871i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: 0.996−0.0871i0.996 - 0.0871i
Analytic conductor: 37.268737.2687
Root analytic conductor: 6.104816.10481
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ162(107,⋅)\chi_{162} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :3), 0.996−0.0871i)(2,\ 162,\ (\ :3),\ 0.996 - 0.0871i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.48225059610.4822505961
L(12)L(\frac12) ≈\approx 0.48225059610.4822505961
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+(4.89+2.82i)T 1 + (4.89 + 2.82i)T
3 1 1
good5 1+(180.−104.i)T+(7.81e3−1.35e4i)T2 1 + (180. - 104. i)T + (7.81e3 - 1.35e4i)T^{2}
7 1+(2.09−3.63i)T+(−5.88e4−1.01e5i)T2 1 + (2.09 - 3.63i)T + (-5.88e4 - 1.01e5i)T^{2}
11 1+(1.95e3+1.13e3i)T+(8.85e5+1.53e6i)T2 1 + (1.95e3 + 1.13e3i)T + (8.85e5 + 1.53e6i)T^{2}
13 1+(1.42e3+2.45e3i)T+(−2.41e6+4.18e6i)T2 1 + (1.42e3 + 2.45e3i)T + (-2.41e6 + 4.18e6i)T^{2}
17 1+1.96e3iT−2.41e7T2 1 + 1.96e3iT - 2.41e7T^{2}
19 1+281.T+4.70e7T2 1 + 281.T + 4.70e7T^{2}
23 1+(1.45e4−8.37e3i)T+(7.40e7−1.28e8i)T2 1 + (1.45e4 - 8.37e3i)T + (7.40e7 - 1.28e8i)T^{2}
29 1+(−3.21e4−1.85e4i)T+(2.97e8+5.15e8i)T2 1 + (-3.21e4 - 1.85e4i)T + (2.97e8 + 5.15e8i)T^{2}
31 1+(−1.23e4−2.13e4i)T+(−4.43e8+7.68e8i)T2 1 + (-1.23e4 - 2.13e4i)T + (-4.43e8 + 7.68e8i)T^{2}
37 1+1.70e4T+2.56e9T2 1 + 1.70e4T + 2.56e9T^{2}
41 1+(1.00e5−5.81e4i)T+(2.37e9−4.11e9i)T2 1 + (1.00e5 - 5.81e4i)T + (2.37e9 - 4.11e9i)T^{2}
43 1+(−1.53e4+2.65e4i)T+(−3.16e9−5.47e9i)T2 1 + (-1.53e4 + 2.65e4i)T + (-3.16e9 - 5.47e9i)T^{2}
47 1+(6.72e4+3.88e4i)T+(5.38e9+9.33e9i)T2 1 + (6.72e4 + 3.88e4i)T + (5.38e9 + 9.33e9i)T^{2}
53 1+1.38e5iT−2.21e10T2 1 + 1.38e5iT - 2.21e10T^{2}
59 1+(−1.32e5+7.64e4i)T+(2.10e10−3.65e10i)T2 1 + (-1.32e5 + 7.64e4i)T + (2.10e10 - 3.65e10i)T^{2}
61 1+(−8.06e3+1.39e4i)T+(−2.57e10−4.46e10i)T2 1 + (-8.06e3 + 1.39e4i)T + (-2.57e10 - 4.46e10i)T^{2}
67 1+(−2.37e5−4.11e5i)T+(−4.52e10+7.83e10i)T2 1 + (-2.37e5 - 4.11e5i)T + (-4.52e10 + 7.83e10i)T^{2}
71 1−1.50e5iT−1.28e11T2 1 - 1.50e5iT - 1.28e11T^{2}
73 1−3.31e5T+1.51e11T2 1 - 3.31e5T + 1.51e11T^{2}
79 1+(−4.48e5+7.76e5i)T+(−1.21e11−2.10e11i)T2 1 + (-4.48e5 + 7.76e5i)T + (-1.21e11 - 2.10e11i)T^{2}
83 1+(−8.18e5−4.72e5i)T+(1.63e11+2.83e11i)T2 1 + (-8.18e5 - 4.72e5i)T + (1.63e11 + 2.83e11i)T^{2}
89 1+7.90e5iT−4.96e11T2 1 + 7.90e5iT - 4.96e11T^{2}
97 1+(6.96e5−1.20e6i)T+(−4.16e11−7.21e11i)T2 1 + (6.96e5 - 1.20e6i)T + (-4.16e11 - 7.21e11i)T^{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

−11.63172722478951559854310520425, −10.66503051696145987182701205252, −10.11635845462393577201045870143, −8.273561320062938590824129167710, −7.929694605775753604340303948545, −6.84333136132732226748401012423, −5.14841912713564858014791993336, −3.45681939014794562178648851978, −2.73196538470891222377130174115, −0.47457504687966383289599796479, 0.39241375109515108283754120049, 2.19924852971607551157292036424, 4.18627239404344035224339477731, 5.03651258072726456916882753466, 6.75213764551627432406953410065, 7.87040197781803990444301036326, 8.314667518564863423156928215001, 9.626269053975290131129138193692, 10.60327149322312741269621021667, 11.92340554756884448097137387681

Graph of the ZZ-function along the critical line