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Properties

Label 2-120-120.59-c3-0-39
Degree 22
Conductor 120120
Sign 0.274+0.961i0.274 + 0.961i
Analytic cond. 7.080227.08022
Root an. cond. 2.660872.66087
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.22 − 1.74i)2-s + (4.99 − 1.43i)3-s + (1.93 + 7.76i)4-s + (−10.2 + 4.55i)5-s + (−13.6 − 5.51i)6-s + 3.90·7-s + (9.22 − 20.6i)8-s + (22.9 − 14.2i)9-s + (30.6 + 7.64i)10-s − 59.1i·11-s + (20.7 + 36.0i)12-s + 63.6·13-s + (−8.70 − 6.80i)14-s + (−44.5 + 37.3i)15-s + (−56.5 + 29.9i)16-s + 69.6·17-s + ⋯
L(s)  = 1  + (−0.787 − 0.615i)2-s + (0.961 − 0.275i)3-s + (0.241 + 0.970i)4-s + (−0.913 + 0.407i)5-s + (−0.926 − 0.375i)6-s + 0.210·7-s + (0.407 − 0.913i)8-s + (0.848 − 0.529i)9-s + (0.970 + 0.241i)10-s − 1.62i·11-s + (0.499 + 0.866i)12-s + 1.35·13-s + (−0.166 − 0.129i)14-s + (−0.765 + 0.642i)15-s + (−0.883 + 0.468i)16-s + 0.993·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.274+0.961i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.274+0.961i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.274+0.961i0.274 + 0.961i
Analytic conductor: 7.080227.08022
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ120(59,⋅)\chi_{120} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :3/2), 0.274+0.961i)(2,\ 120,\ (\ :3/2),\ 0.274 + 0.961i)

Particular Values

L(2)L(2) ≈\approx 1.11611−0.841709i1.11611 - 0.841709i
L(12)L(\frac12) ≈\approx 1.11611−0.841709i1.11611 - 0.841709i
L(52)L(\frac{5}{2}) 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+(2.22+1.74i)T 1 + (2.22 + 1.74i)T
3 1+(−4.99+1.43i)T 1 + (-4.99 + 1.43i)T
5 1+(10.2−4.55i)T 1 + (10.2 - 4.55i)T
good7 1−3.90T+343T2 1 - 3.90T + 343T^{2}
11 1+59.1iT−1.33e3T2 1 + 59.1iT - 1.33e3T^{2}
13 1−63.6T+2.19e3T2 1 - 63.6T + 2.19e3T^{2}
17 1−69.6T+4.91e3T2 1 - 69.6T + 4.91e3T^{2}
19 1−33.0T+6.85e3T2 1 - 33.0T + 6.85e3T^{2}
23 1+90.6iT−1.21e4T2 1 + 90.6iT - 1.21e4T^{2}
29 1+172.T+2.43e4T2 1 + 172.T + 2.43e4T^{2}
31 1+61.7iT−2.97e4T2 1 + 61.7iT - 2.97e4T^{2}
37 1−10.7T+5.06e4T2 1 - 10.7T + 5.06e4T^{2}
41 1−475.iT−6.89e4T2 1 - 475. iT - 6.89e4T^{2}
43 1+59.3iT−7.95e4T2 1 + 59.3iT - 7.95e4T^{2}
47 1−500.iT−1.03e5T2 1 - 500. iT - 1.03e5T^{2}
53 1+407.iT−1.48e5T2 1 + 407. iT - 1.48e5T^{2}
59 1+17.5iT−2.05e5T2 1 + 17.5iT - 2.05e5T^{2}
61 1+245.iT−2.26e5T2 1 + 245. iT - 2.26e5T^{2}
67 1+35.7iT−3.00e5T2 1 + 35.7iT - 3.00e5T^{2}
71 1−889.T+3.57e5T2 1 - 889.T + 3.57e5T^{2}
73 1−617.iT−3.89e5T2 1 - 617. iT - 3.89e5T^{2}
79 1+108.iT−4.93e5T2 1 + 108. iT - 4.93e5T^{2}
83 1−628.T+5.71e5T2 1 - 628.T + 5.71e5T^{2}
89 1−763.iT−7.04e5T2 1 - 763. iT - 7.04e5T^{2}
97 1−866.iT−9.12e5T2 1 - 866. iT - 9.12e5T^{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

−12.71679568278368720517651578166, −11.47467633593362322027855479364, −10.85585771542719559912010718294, −9.455992026320920378205026907691, −8.252175287434550092908911758023, −7.982991884679194434209737880894, −6.48804389793244257029926575495, −3.79506748721267942519844160594, −3.05070279986755862695519478143, −1.01200754009842438148616136250, 1.57106548915839942776122218170, 3.80572968340200023170056499490, 5.20489421087130044160239080381, 7.18181312590591652066989425360, 7.83572260521266257542564960635, 8.841816852474679430827210854380, 9.715842443971129840650637546424, 10.83413012643085445710599155825, 12.13481544873146718119826027817, 13.44497706955114544141324777027

Graph of the ZZ-function along the critical line