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Properties

Label 2-120-120.59-c3-0-53
Degree 22
Conductor 120120
Sign −0.793+0.609i-0.793 + 0.609i
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  + (−2.80 − 0.363i)2-s + (3.73 + 3.61i)3-s + (7.73 + 2.03i)4-s + (−2.66 − 10.8i)5-s + (−9.15 − 11.4i)6-s − 26.3·7-s + (−20.9 − 8.53i)8-s + (0.853 + 26.9i)9-s + (3.52 + 31.4i)10-s − 37.4i·11-s + (21.4 + 35.5i)12-s − 30.8·13-s + (73.8 + 9.57i)14-s + (29.3 − 50.1i)15-s + (55.6 + 31.5i)16-s − 54.0·17-s + ⋯
L(s)  = 1  + (−0.991 − 0.128i)2-s + (0.718 + 0.695i)3-s + (0.966 + 0.254i)4-s + (−0.238 − 0.971i)5-s + (−0.622 − 0.782i)6-s − 1.42·7-s + (−0.926 − 0.377i)8-s + (0.0315 + 0.999i)9-s + (0.111 + 0.993i)10-s − 1.02i·11-s + (0.517 + 0.855i)12-s − 0.658·13-s + (1.41 + 0.182i)14-s + (0.504 − 0.863i)15-s + (0.869 + 0.493i)16-s − 0.770·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((−0.793+0.609i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.609i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((−0.793+0.609i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −0.793+0.609i-0.793 + 0.609i
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), −0.793+0.609i)(2,\ 120,\ (\ :3/2),\ -0.793 + 0.609i)

Particular Values

L(2)L(2) ≈\approx 0.107287−0.315801i0.107287 - 0.315801i
L(12)L(\frac12) ≈\approx 0.107287−0.315801i0.107287 - 0.315801i
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+(2.80+0.363i)T 1 + (2.80 + 0.363i)T
3 1+(−3.73−3.61i)T 1 + (-3.73 - 3.61i)T
5 1+(2.66+10.8i)T 1 + (2.66 + 10.8i)T
good7 1+26.3T+343T2 1 + 26.3T + 343T^{2}
11 1+37.4iT−1.33e3T2 1 + 37.4iT - 1.33e3T^{2}
13 1+30.8T+2.19e3T2 1 + 30.8T + 2.19e3T^{2}
17 1+54.0T+4.91e3T2 1 + 54.0T + 4.91e3T^{2}
19 1+4.70T+6.85e3T2 1 + 4.70T + 6.85e3T^{2}
23 1+129.iT−1.21e4T2 1 + 129. iT - 1.21e4T^{2}
29 1+230.T+2.43e4T2 1 + 230.T + 2.43e4T^{2}
31 1+123.iT−2.97e4T2 1 + 123. iT - 2.97e4T^{2}
37 1−349.T+5.06e4T2 1 - 349.T + 5.06e4T^{2}
41 1+74.8iT−6.89e4T2 1 + 74.8iT - 6.89e4T^{2}
43 1+364.iT−7.95e4T2 1 + 364. iT - 7.95e4T^{2}
47 1−45.7iT−1.03e5T2 1 - 45.7iT - 1.03e5T^{2}
53 1−682.iT−1.48e5T2 1 - 682. iT - 1.48e5T^{2}
59 1+256.iT−2.05e5T2 1 + 256. iT - 2.05e5T^{2}
61 1−435.iT−2.26e5T2 1 - 435. iT - 2.26e5T^{2}
67 1−862.iT−3.00e5T2 1 - 862. iT - 3.00e5T^{2}
71 1+366.T+3.57e5T2 1 + 366.T + 3.57e5T^{2}
73 1−215.iT−3.89e5T2 1 - 215. iT - 3.89e5T^{2}
79 1+340.iT−4.93e5T2 1 + 340. iT - 4.93e5T^{2}
83 1−605.T+5.71e5T2 1 - 605.T + 5.71e5T^{2}
89 1+517.iT−7.04e5T2 1 + 517. iT - 7.04e5T^{2}
97 1+1.44e3iT−9.12e5T2 1 + 1.44e3iT - 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

−12.62516453249140173841957261453, −11.28737798667626105595838983620, −10.14704323953590185144443593507, −9.239323801054179372515518876786, −8.694304580940633455214773014814, −7.47738882025109264166447331254, −5.91999431233180023122357052053, −3.99395356469170939105476339018, −2.62162550851288171926658322204, −0.20185989247542140925974150557, 2.19734199452398271760378449969, 3.35949440418777668274287292707, 6.30333512571246476411855801100, 7.04476492639171968270027678413, 7.79639071143960963788572039936, 9.428988937544831428250940994298, 9.781303622328604896882767054128, 11.22997690064226830232576704341, 12.36618863162219233389018709150, 13.30829355614916009717298802764

Graph of the ZZ-function along the critical line