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Properties

Label 2-120-120.53-c3-0-39
Degree 22
Conductor 120120
Sign 0.988+0.154i0.988 + 0.154i
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  + (−0.401 + 2.79i)2-s + (−2.36 + 4.62i)3-s + (−7.67 − 2.24i)4-s + (3.00 − 10.7i)5-s + (−12.0 − 8.48i)6-s + (13.4 − 13.4i)7-s + (9.37 − 20.5i)8-s + (−15.8 − 21.8i)9-s + (28.9 + 12.7i)10-s − 30.2·11-s + (28.5 − 30.2i)12-s + (25.9 − 25.9i)13-s + (32.2 + 43.0i)14-s + (42.7 + 39.3i)15-s + (53.8 + 34.5i)16-s + (−26.6 − 26.6i)17-s + ⋯
L(s)  = 1  + (−0.141 + 0.989i)2-s + (−0.455 + 0.890i)3-s + (−0.959 − 0.281i)4-s + (0.268 − 0.963i)5-s + (−0.816 − 0.577i)6-s + (0.725 − 0.725i)7-s + (0.414 − 0.910i)8-s + (−0.585 − 0.810i)9-s + (0.915 + 0.402i)10-s − 0.829·11-s + (0.687 − 0.726i)12-s + (0.553 − 0.553i)13-s + (0.614 + 0.820i)14-s + (0.735 + 0.677i)15-s + (0.841 + 0.539i)16-s + (−0.380 − 0.380i)17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.988+0.154i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.988+0.154i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.988+0.154i0.988 + 0.154i
Analytic conductor: 7.080227.08022
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ120(53,⋅)\chi_{120} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :3/2), 0.988+0.154i)(2,\ 120,\ (\ :3/2),\ 0.988 + 0.154i)

Particular Values

L(2)L(2) ≈\approx 1.00132−0.0776762i1.00132 - 0.0776762i
L(12)L(\frac12) ≈\approx 1.00132−0.0776762i1.00132 - 0.0776762i
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+(0.401−2.79i)T 1 + (0.401 - 2.79i)T
3 1+(2.36−4.62i)T 1 + (2.36 - 4.62i)T
5 1+(−3.00+10.7i)T 1 + (-3.00 + 10.7i)T
good7 1+(−13.4+13.4i)T−343iT2 1 + (-13.4 + 13.4i)T - 343iT^{2}
11 1+30.2T+1.33e3T2 1 + 30.2T + 1.33e3T^{2}
13 1+(−25.9+25.9i)T−2.19e3iT2 1 + (-25.9 + 25.9i)T - 2.19e3iT^{2}
17 1+(26.6+26.6i)T+4.91e3iT2 1 + (26.6 + 26.6i)T + 4.91e3iT^{2}
19 1−48.6T+6.85e3T2 1 - 48.6T + 6.85e3T^{2}
23 1+(59.4−59.4i)T−1.21e4iT2 1 + (59.4 - 59.4i)T - 1.21e4iT^{2}
29 1+254.iT−2.43e4T2 1 + 254. iT - 2.43e4T^{2}
31 1−240.T+2.97e4T2 1 - 240.T + 2.97e4T^{2}
37 1+(177.+177.i)T+5.06e4iT2 1 + (177. + 177. i)T + 5.06e4iT^{2}
41 1−88.8iT−6.89e4T2 1 - 88.8iT - 6.89e4T^{2}
43 1+(−335.+335.i)T−7.95e4iT2 1 + (-335. + 335. i)T - 7.95e4iT^{2}
47 1+(316.+316.i)T+1.03e5iT2 1 + (316. + 316. i)T + 1.03e5iT^{2}
53 1+(−53.4−53.4i)T+1.48e5iT2 1 + (-53.4 - 53.4i)T + 1.48e5iT^{2}
59 1−26.0iT−2.05e5T2 1 - 26.0iT - 2.05e5T^{2}
61 1−685.iT−2.26e5T2 1 - 685. iT - 2.26e5T^{2}
67 1+(429.+429.i)T+3.00e5iT2 1 + (429. + 429. i)T + 3.00e5iT^{2}
71 1−2.66iT−3.57e5T2 1 - 2.66iT - 3.57e5T^{2}
73 1+(417.+417.i)T+3.89e5iT2 1 + (417. + 417. i)T + 3.89e5iT^{2}
79 1−662.iT−4.93e5T2 1 - 662. iT - 4.93e5T^{2}
83 1+(−915.−915.i)T+5.71e5iT2 1 + (-915. - 915. i)T + 5.71e5iT^{2}
89 1+470.T+7.04e5T2 1 + 470.T + 7.04e5T^{2}
97 1+(−1.20e3+1.20e3i)T−9.12e5iT2 1 + (-1.20e3 + 1.20e3i)T - 9.12e5iT^{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

−13.37380165393008807997187366122, −11.92044008621227347848880714622, −10.56659475666374944000192864059, −9.742626145513208592919144320902, −8.600208441192221651434222512809, −7.67556532289017712176786447784, −5.93610327235817686342233306186, −5.06189965554009759426638603041, −4.10843056656529638414291803472, −0.62116183402971448224425973737, 1.69318367658863934971517454210, 2.86297884231410036436295397140, 4.97807956183264880571556483675, 6.26683185797364845688477754988, 7.74613020052404992431146502098, 8.721458172591559297055762308116, 10.28341194029325590778427553065, 11.11510483144459059863582917891, 11.81735957079665405820898037727, 12.84818726797171415670495408168

Graph of the ZZ-function along the critical line