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Properties

Label 2-120-120.59-c3-0-30
Degree 22
Conductor 120120
Sign 0.819+0.573i0.819 + 0.573i
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  + (−2.73 + 0.732i)2-s + (−2.64 − 4.47i)3-s + (6.92 − 4.00i)4-s + (10.4 + 3.92i)5-s + (10.4 + 10.2i)6-s + 4.10·7-s + (−15.9 + 16.0i)8-s + (−13.0 + 23.6i)9-s + (−31.4 − 3.06i)10-s − 5.96i·11-s + (−36.2 − 20.4i)12-s + 33.0·13-s + (−11.2 + 3.00i)14-s + (−10.0 − 57.2i)15-s + (31.9 − 55.4i)16-s + 50.6·17-s + ⋯
L(s)  = 1  + (−0.965 + 0.259i)2-s + (−0.508 − 0.861i)3-s + (0.865 − 0.500i)4-s + (0.936 + 0.351i)5-s + (0.714 + 0.699i)6-s + 0.221·7-s + (−0.706 + 0.707i)8-s + (−0.483 + 0.875i)9-s + (−0.995 − 0.0968i)10-s − 0.163i·11-s + (−0.871 − 0.491i)12-s + 0.705·13-s + (−0.213 + 0.0573i)14-s + (−0.173 − 0.984i)15-s + (0.498 − 0.866i)16-s + 0.722·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.819+0.573i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.819+0.573i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.819+0.573i0.819 + 0.573i
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), 0.819+0.573i)(2,\ 120,\ (\ :3/2),\ 0.819 + 0.573i)

Particular Values

L(2)L(2) ≈\approx 1.03344−0.325511i1.03344 - 0.325511i
L(12)L(\frac12) ≈\approx 1.03344−0.325511i1.03344 - 0.325511i
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+(2.73−0.732i)T 1 + (2.73 - 0.732i)T
3 1+(2.64+4.47i)T 1 + (2.64 + 4.47i)T
5 1+(−10.4−3.92i)T 1 + (-10.4 - 3.92i)T
good7 1−4.10T+343T2 1 - 4.10T + 343T^{2}
11 1+5.96iT−1.33e3T2 1 + 5.96iT - 1.33e3T^{2}
13 1−33.0T+2.19e3T2 1 - 33.0T + 2.19e3T^{2}
17 1−50.6T+4.91e3T2 1 - 50.6T + 4.91e3T^{2}
19 1−74.1T+6.85e3T2 1 - 74.1T + 6.85e3T^{2}
23 1+184.iT−1.21e4T2 1 + 184. iT - 1.21e4T^{2}
29 1+98.3T+2.43e4T2 1 + 98.3T + 2.43e4T^{2}
31 1+192.iT−2.97e4T2 1 + 192. iT - 2.97e4T^{2}
37 1−350.T+5.06e4T2 1 - 350.T + 5.06e4T^{2}
41 1+292.iT−6.89e4T2 1 + 292. iT - 6.89e4T^{2}
43 1−80.8iT−7.95e4T2 1 - 80.8iT - 7.95e4T^{2}
47 1+63.1iT−1.03e5T2 1 + 63.1iT - 1.03e5T^{2}
53 1+178.iT−1.48e5T2 1 + 178. iT - 1.48e5T^{2}
59 1−479.iT−2.05e5T2 1 - 479. iT - 2.05e5T^{2}
61 1−635.iT−2.26e5T2 1 - 635. iT - 2.26e5T^{2}
67 1−288.iT−3.00e5T2 1 - 288. iT - 3.00e5T^{2}
71 1−1.06e3T+3.57e5T2 1 - 1.06e3T + 3.57e5T^{2}
73 1−980.iT−3.89e5T2 1 - 980. iT - 3.89e5T^{2}
79 1+804.iT−4.93e5T2 1 + 804. iT - 4.93e5T^{2}
83 1+300.T+5.71e5T2 1 + 300.T + 5.71e5T^{2}
89 1−1.11e3iT−7.04e5T2 1 - 1.11e3iT - 7.04e5T^{2}
97 1+1.21e3iT−9.12e5T2 1 + 1.21e3iT - 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

−12.86133507144015835753011783058, −11.59054176470156419456473369895, −10.78441107859683661267410107251, −9.759452689137890556765908111473, −8.487154224067568933786365349409, −7.40749976938781413663367423877, −6.31215640547608428652997393740, −5.53472971034587753049075188206, −2.47408219385506604658810368051, −1.02980194465050749378002030977, 1.29386079228892506296343253379, 3.36130179326631872327721562395, 5.25227002745845170794244817513, 6.33901104706569913996485301887, 7.949601423614781140633791711954, 9.332743087569820196407024533259, 9.722797264566070273995417260856, 10.87468671666714076527953951681, 11.68214318980568211859748274218, 12.84938008804796850199110568112

Graph of the ZZ-function along the critical line