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Properties

Label 2-120-120.59-c3-0-18
Degree 22
Conductor 120120
Sign 0.983−0.181i0.983 - 0.181i
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  + (−1.65 + 2.29i)2-s + (−4.82 − 1.93i)3-s + (−2.51 − 7.59i)4-s + (−7.20 + 8.54i)5-s + (12.4 − 7.85i)6-s − 30.0·7-s + (21.5 + 6.82i)8-s + (19.5 + 18.6i)9-s + (−7.65 − 30.6i)10-s − 35.6i·11-s + (−2.55 + 41.4i)12-s + 58.6·13-s + (49.7 − 68.8i)14-s + (51.2 − 27.3i)15-s + (−51.3 + 38.1i)16-s − 14.3·17-s + ⋯
L(s)  = 1  + (−0.585 + 0.810i)2-s + (−0.928 − 0.371i)3-s + (−0.313 − 0.949i)4-s + (−0.644 + 0.764i)5-s + (0.845 − 0.534i)6-s − 1.62·7-s + (0.953 + 0.301i)8-s + (0.723 + 0.690i)9-s + (−0.242 − 0.970i)10-s − 0.976i·11-s + (−0.0615 + 0.998i)12-s + 1.25·13-s + (0.950 − 1.31i)14-s + (0.882 − 0.470i)15-s + (−0.803 + 0.595i)16-s − 0.205·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.983−0.181i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.983−0.181i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.983−0.181i0.983 - 0.181i
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), 0.983−0.181i)(2,\ 120,\ (\ :3/2),\ 0.983 - 0.181i)

Particular Values

L(2)L(2) ≈\approx 0.528058+0.0484124i0.528058 + 0.0484124i
L(12)L(\frac12) ≈\approx 0.528058+0.0484124i0.528058 + 0.0484124i
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+(1.65−2.29i)T 1 + (1.65 - 2.29i)T
3 1+(4.82+1.93i)T 1 + (4.82 + 1.93i)T
5 1+(7.20−8.54i)T 1 + (7.20 - 8.54i)T
good7 1+30.0T+343T2 1 + 30.0T + 343T^{2}
11 1+35.6iT−1.33e3T2 1 + 35.6iT - 1.33e3T^{2}
13 1−58.6T+2.19e3T2 1 - 58.6T + 2.19e3T^{2}
17 1+14.3T+4.91e3T2 1 + 14.3T + 4.91e3T^{2}
19 1−106.T+6.85e3T2 1 - 106.T + 6.85e3T^{2}
23 1−47.7iT−1.21e4T2 1 - 47.7iT - 1.21e4T^{2}
29 1−49.5T+2.43e4T2 1 - 49.5T + 2.43e4T^{2}
31 1−181.iT−2.97e4T2 1 - 181. iT - 2.97e4T^{2}
37 1−113.T+5.06e4T2 1 - 113.T + 5.06e4T^{2}
41 1−53.6iT−6.89e4T2 1 - 53.6iT - 6.89e4T^{2}
43 1+490.iT−7.95e4T2 1 + 490. iT - 7.95e4T^{2}
47 1+441.iT−1.03e5T2 1 + 441. iT - 1.03e5T^{2}
53 1+65.6iT−1.48e5T2 1 + 65.6iT - 1.48e5T^{2}
59 1+406.iT−2.05e5T2 1 + 406. iT - 2.05e5T^{2}
61 1−213.iT−2.26e5T2 1 - 213. iT - 2.26e5T^{2}
67 1−825.iT−3.00e5T2 1 - 825. iT - 3.00e5T^{2}
71 1−157.T+3.57e5T2 1 - 157.T + 3.57e5T^{2}
73 1−242.iT−3.89e5T2 1 - 242. iT - 3.89e5T^{2}
79 1−107.iT−4.93e5T2 1 - 107. iT - 4.93e5T^{2}
83 1−1.15e3T+5.71e5T2 1 - 1.15e3T + 5.71e5T^{2}
89 1+750.iT−7.04e5T2 1 + 750. iT - 7.04e5T^{2}
97 1+83.2iT−9.12e5T2 1 + 83.2iT - 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

−13.24149488446692525184115346236, −11.78162359550188760334390324906, −10.82209424358109925379038392889, −9.972763876660120861080732307689, −8.590285000359589609247597982174, −7.22593381605970798868260280741, −6.49268083037701251169662602749, −5.64231614543152652089393671086, −3.54487127250043970902043184477, −0.59675902356634427387737661191, 0.849518751891917636464977430070, 3.44572051937694015522049235705, 4.54231963217077823280862609900, 6.28957770951676946223575063374, 7.62827230000392002197237700592, 9.223594630287705607433450124150, 9.737015455660715415267895532556, 10.93147736585823441157586337263, 11.90838279775828522299163628135, 12.67027076261770106514884958124

Graph of the ZZ-function along the critical line