↑

Properties

Label 2-120-120.53-c3-0-31
Degree 22
Conductor 120120
Sign 0.856−0.515i0.856 - 0.515i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.75 + 0.631i)2-s + (−4.07 − 3.22i)3-s + (7.20 + 3.48i)4-s + (8.53 + 7.21i)5-s + (−9.20 − 11.4i)6-s + (−9.44 + 9.44i)7-s + (17.6 + 14.1i)8-s + (6.24 + 26.2i)9-s + (18.9 + 25.2i)10-s + 41.6·11-s + (−18.1 − 37.4i)12-s + (53.3 − 53.3i)13-s + (−32.0 + 20.0i)14-s + (−11.5 − 56.9i)15-s + (39.7 + 50.1i)16-s + (41.4 + 41.4i)17-s + ⋯
L(s)  = 1  + (0.974 + 0.223i)2-s + (−0.784 − 0.620i)3-s + (0.900 + 0.435i)4-s + (0.763 + 0.645i)5-s + (−0.626 − 0.779i)6-s + (−0.510 + 0.510i)7-s + (0.780 + 0.625i)8-s + (0.231 + 0.972i)9-s + (0.600 + 0.799i)10-s + 1.14·11-s + (−0.436 − 0.899i)12-s + (1.13 − 1.13i)13-s + (−0.611 + 0.383i)14-s + (−0.198 − 0.980i)15-s + (0.620 + 0.784i)16-s + (0.591 + 0.591i)17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.856−0.515i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.856−0.515i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.856−0.515i0.856 - 0.515i
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), 0.856−0.515i)(2,\ 120,\ (\ :3/2),\ 0.856 - 0.515i)

Particular Values

L(2)L(2) ≈\approx 2.51219+0.697319i2.51219 + 0.697319i
L(12)L(\frac12) ≈\approx 2.51219+0.697319i2.51219 + 0.697319i
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+(−2.75−0.631i)T 1 + (-2.75 - 0.631i)T
3 1+(4.07+3.22i)T 1 + (4.07 + 3.22i)T
5 1+(−8.53−7.21i)T 1 + (-8.53 - 7.21i)T
good7 1+(9.44−9.44i)T−343iT2 1 + (9.44 - 9.44i)T - 343iT^{2}
11 1−41.6T+1.33e3T2 1 - 41.6T + 1.33e3T^{2}
13 1+(−53.3+53.3i)T−2.19e3iT2 1 + (-53.3 + 53.3i)T - 2.19e3iT^{2}
17 1+(−41.4−41.4i)T+4.91e3iT2 1 + (-41.4 - 41.4i)T + 4.91e3iT^{2}
19 1+145.T+6.85e3T2 1 + 145.T + 6.85e3T^{2}
23 1+(−27.2+27.2i)T−1.21e4iT2 1 + (-27.2 + 27.2i)T - 1.21e4iT^{2}
29 1−107.iT−2.43e4T2 1 - 107. iT - 2.43e4T^{2}
31 1+151.T+2.97e4T2 1 + 151.T + 2.97e4T^{2}
37 1+(284.+284.i)T+5.06e4iT2 1 + (284. + 284. i)T + 5.06e4iT^{2}
41 1+203.iT−6.89e4T2 1 + 203. iT - 6.89e4T^{2}
43 1+(−194.+194.i)T−7.95e4iT2 1 + (-194. + 194. i)T - 7.95e4iT^{2}
47 1+(197.+197.i)T+1.03e5iT2 1 + (197. + 197. i)T + 1.03e5iT^{2}
53 1+(202.+202.i)T+1.48e5iT2 1 + (202. + 202. i)T + 1.48e5iT^{2}
59 1+460.iT−2.05e5T2 1 + 460. iT - 2.05e5T^{2}
61 1−94.5iT−2.26e5T2 1 - 94.5iT - 2.26e5T^{2}
67 1+(−258.−258.i)T+3.00e5iT2 1 + (-258. - 258. i)T + 3.00e5iT^{2}
71 1+76.8iT−3.57e5T2 1 + 76.8iT - 3.57e5T^{2}
73 1+(−14.0−14.0i)T+3.89e5iT2 1 + (-14.0 - 14.0i)T + 3.89e5iT^{2}
79 1−221.iT−4.93e5T2 1 - 221. iT - 4.93e5T^{2}
83 1+(38.4+38.4i)T+5.71e5iT2 1 + (38.4 + 38.4i)T + 5.71e5iT^{2}
89 1+230.T+7.04e5T2 1 + 230.T + 7.04e5T^{2}
97 1+(−758.+758.i)T−9.12e5iT2 1 + (-758. + 758. i)T - 9.12e5iT^{2}
show more
show less
   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

−12.83497954889581214214004164013, −12.52890147747600200333807332008, −11.06998396097544548602854243494, −10.49991148461570502667701101434, −8.600956317551280758748169977073, −7.00698589514958921732367987330, −6.18801212394933060224684548746, −5.56085316854767039780684320393, −3.59083252447403318562939412125, −1.89143592833242456771719557399, 1.36610216087989832299820429916, 3.77036271635818135562122281891, 4.67755381829312107188720441209, 6.12817145255476764408466081484, 6.62240346253192063301188635000, 9.033454260657084619798557875227, 9.961610717092506640544663989415, 11.02903033371589609284762008626, 11.91672281736447066750399824000, 12.90920310871747642349251331801

Graph of the ZZ-function along the critical line