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Properties

Label 2-120-5.4-c3-0-5
Degree 22
Conductor 120120
Sign −0.0160+0.999i-0.0160 + 0.999i
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  − 3i·3-s + (−0.178 + 11.1i)5-s − 35.0i·7-s − 9·9-s + 25.6·11-s − 37.6i·13-s + (33.5 + 0.536i)15-s − 95.7i·17-s − 50.8·19-s − 105.·21-s − 110. i·23-s + (−124. − 4i)25-s + 27i·27-s + 54.5·29-s + 198.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.0160 + 0.999i)5-s − 1.89i·7-s − 0.333·9-s + 0.702·11-s − 0.803i·13-s + (0.577 + 0.00923i)15-s − 1.36i·17-s − 0.614·19-s − 1.09·21-s − 1.00i·23-s + (−0.999 − 0.0320i)25-s + 0.192i·27-s + 0.349·29-s + 1.14·31-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((−0.0160+0.999i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0160 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((−0.0160+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0160 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −0.0160+0.999i-0.0160 + 0.999i
Analytic conductor: 7.080227.08022
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ120(49,⋅)\chi_{120} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :3/2), −0.0160+0.999i)(2,\ 120,\ (\ :3/2),\ -0.0160 + 0.999i)

Particular Values

L(2)L(2) ≈\approx 1.00016−1.01629i1.00016 - 1.01629i
L(12)L(\frac12) ≈\approx 1.00016−1.01629i1.00016 - 1.01629i
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
3 1+3iT 1 + 3iT
5 1+(0.178−11.1i)T 1 + (0.178 - 11.1i)T
good7 1+35.0iT−343T2 1 + 35.0iT - 343T^{2}
11 1−25.6T+1.33e3T2 1 - 25.6T + 1.33e3T^{2}
13 1+37.6iT−2.19e3T2 1 + 37.6iT - 2.19e3T^{2}
17 1+95.7iT−4.91e3T2 1 + 95.7iT - 4.91e3T^{2}
19 1+50.8T+6.85e3T2 1 + 50.8T + 6.85e3T^{2}
23 1+110.iT−1.21e4T2 1 + 110. iT - 1.21e4T^{2}
29 1−54.5T+2.43e4T2 1 - 54.5T + 2.43e4T^{2}
31 1−198.T+2.97e4T2 1 - 198.T + 2.97e4T^{2}
37 1−266.iT−5.06e4T2 1 - 266. iT - 5.06e4T^{2}
41 1−103.T+6.89e4T2 1 - 103.T + 6.89e4T^{2}
43 1+108iT−7.95e4T2 1 + 108iT - 7.95e4T^{2}
47 1−597.iT−1.03e5T2 1 - 597. iT - 1.03e5T^{2}
53 1−305.iT−1.48e5T2 1 - 305. iT - 1.48e5T^{2}
59 1−223.T+2.05e5T2 1 - 223.T + 2.05e5T^{2}
61 1−485.T+2.26e5T2 1 - 485.T + 2.26e5T^{2}
67 1+876.iT−3.00e5T2 1 + 876. iT - 3.00e5T^{2}
71 1−585.T+3.57e5T2 1 - 585.T + 3.57e5T^{2}
73 1−1.13e3iT−3.89e5T2 1 - 1.13e3iT - 3.89e5T^{2}
79 1+685.T+4.93e5T2 1 + 685.T + 4.93e5T^{2}
83 1+305.iT−5.71e5T2 1 + 305. iT - 5.71e5T^{2}
89 1+887.T+7.04e5T2 1 + 887.T + 7.04e5T^{2}
97 1+556.iT−9.12e5T2 1 + 556. 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

−12.90145817088641617253109624584, −11.56167035025881223762813448658, −10.67781277471578965879239303960, −9.856823163376597715731293943724, −8.075174455443012467585554249264, −7.11271283942209307565796333633, −6.43602118167436456770529261870, −4.38897258711049096191606133481, −2.95304287587602149070490726955, −0.78487325745986751004655743653, 1.98614592785421455403547198093, 3.98283864078735772869535659546, 5.29910222218419789012278081000, 6.24877512724909560348703093536, 8.404399285178460973213610555870, 8.905456424066426732818790223438, 9.831280694064448654710245180772, 11.48527701061669095071560294147, 12.11807439073390823992758252608, 13.05192109501944838396503415050

Graph of the ZZ-function along the critical line