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Properties

Label 2-162-3.2-c6-0-13
Degree 22
Conductor 162162
Sign ii
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 + 24.8i·5-s − 6.19·7-s + 181. i·8-s + 140.·10-s − 130. i·11-s − 922.·13-s + 35.0i·14-s + 1.02e3·16-s + 3.38e3i·17-s + 5.40e3·19-s − 794. i·20-s − 740.·22-s − 2.36e3i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 0.198i·5-s − 0.0180·7-s + 0.353i·8-s + 0.140·10-s − 0.0983i·11-s − 0.419·13-s + 0.0127i·14-s + 0.250·16-s + 0.688i·17-s + 0.787·19-s − 0.0992i·20-s − 0.0695·22-s − 0.194i·23-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: ii
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), i)(2,\ 162,\ (\ :3),\ i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.6669573471.666957347
L(12)L(\frac12) ≈\approx 1.6669573471.666957347
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 1−24.8iT−1.56e4T2 1 - 24.8iT - 1.56e4T^{2}
7 1+6.19T+1.17e5T2 1 + 6.19T + 1.17e5T^{2}
11 1+130.iT−1.77e6T2 1 + 130. iT - 1.77e6T^{2}
13 1+922.T+4.82e6T2 1 + 922.T + 4.82e6T^{2}
17 1−3.38e3iT−2.41e7T2 1 - 3.38e3iT - 2.41e7T^{2}
19 1−5.40e3T+4.70e7T2 1 - 5.40e3T + 4.70e7T^{2}
23 1+2.36e3iT−1.48e8T2 1 + 2.36e3iT - 1.48e8T^{2}
29 1+3.49e4iT−5.94e8T2 1 + 3.49e4iT - 5.94e8T^{2}
31 1−4.66e3T+8.87e8T2 1 - 4.66e3T + 8.87e8T^{2}
37 1−6.91e3T+2.56e9T2 1 - 6.91e3T + 2.56e9T^{2}
41 1+5.38e4iT−4.75e9T2 1 + 5.38e4iT - 4.75e9T^{2}
43 1−1.23e5T+6.32e9T2 1 - 1.23e5T + 6.32e9T^{2}
47 1+9.60e4iT−1.07e10T2 1 + 9.60e4iT - 1.07e10T^{2}
53 1+1.32e5iT−2.21e10T2 1 + 1.32e5iT - 2.21e10T^{2}
59 1+2.90e4iT−4.21e10T2 1 + 2.90e4iT - 4.21e10T^{2}
61 1−3.20e5T+5.15e10T2 1 - 3.20e5T + 5.15e10T^{2}
67 1+5.29e5T+9.04e10T2 1 + 5.29e5T + 9.04e10T^{2}
71 1+6.28e5iT−1.28e11T2 1 + 6.28e5iT - 1.28e11T^{2}
73 1+6.87e4T+1.51e11T2 1 + 6.87e4T + 1.51e11T^{2}
79 1−4.71e5T+2.43e11T2 1 - 4.71e5T + 2.43e11T^{2}
83 1+3.35e4iT−3.26e11T2 1 + 3.35e4iT - 3.26e11T^{2}
89 1+1.01e6iT−4.96e11T2 1 + 1.01e6iT - 4.96e11T^{2}
97 1−4.18e5T+8.32e11T2 1 - 4.18e5T + 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

−11.50318721116501458078081881933, −10.52959532295513050359998844929, −9.662541171565286228910613855795, −8.564081670731083971015383943257, −7.40765341690456207880144272247, −6.01337324632194373119299402676, −4.70088089819609813924858913189, −3.41748306930860739481483671933, −2.15065574576836204074426493832, −0.60825224092598202943167468638, 1.02765213242165244982656942075, 2.97299123776464371834841582617, 4.55643415361811843409152634271, 5.51357133450215452671542029177, 6.83792085026262081554111260614, 7.69911966681077427095742372652, 8.909182436333743728544929509819, 9.721681722110058819379387556930, 10.97009340293895737513523714509, 12.17273286561426335234510804519

Graph of the ZZ-function along the critical line