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Properties

Label 2-120-120.53-c3-0-22
Degree 22
Conductor 120120
Sign 0.882+0.469i0.882 + 0.469i
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  + (−2.82 − 0.0740i)2-s + (−4.92 − 1.65i)3-s + (7.98 + 0.418i)4-s + (9.98 − 5.02i)5-s + (13.8 + 5.03i)6-s + (−17.9 + 17.9i)7-s + (−22.5 − 1.77i)8-s + (21.5 + 16.2i)9-s + (−28.6 + 13.4i)10-s − 22.2·11-s + (−38.6 − 15.2i)12-s + (31.5 − 31.5i)13-s + (51.9 − 49.3i)14-s + (−57.5 + 8.27i)15-s + (63.6 + 6.68i)16-s + (17.9 + 17.9i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0261i)2-s + (−0.948 − 0.317i)3-s + (0.998 + 0.0523i)4-s + (0.893 − 0.449i)5-s + (0.939 + 0.342i)6-s + (−0.967 + 0.967i)7-s + (−0.996 − 0.0784i)8-s + (0.798 + 0.602i)9-s + (−0.904 + 0.425i)10-s − 0.609·11-s + (−0.930 − 0.366i)12-s + (0.673 − 0.673i)13-s + (0.992 − 0.941i)14-s + (−0.989 + 0.142i)15-s + (0.994 + 0.104i)16-s + (0.255 + 0.255i)17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.882+0.469i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.882+0.469i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.882+0.469i0.882 + 0.469i
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), 0.882+0.469i)(2,\ 120,\ (\ :3/2),\ 0.882 + 0.469i)

Particular Values

L(2)L(2) ≈\approx 0.777751−0.194148i0.777751 - 0.194148i
L(12)L(\frac12) ≈\approx 0.777751−0.194148i0.777751 - 0.194148i
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+(2.82+0.0740i)T 1 + (2.82 + 0.0740i)T
3 1+(4.92+1.65i)T 1 + (4.92 + 1.65i)T
5 1+(−9.98+5.02i)T 1 + (-9.98 + 5.02i)T
good7 1+(17.9−17.9i)T−343iT2 1 + (17.9 - 17.9i)T - 343iT^{2}
11 1+22.2T+1.33e3T2 1 + 22.2T + 1.33e3T^{2}
13 1+(−31.5+31.5i)T−2.19e3iT2 1 + (-31.5 + 31.5i)T - 2.19e3iT^{2}
17 1+(−17.9−17.9i)T+4.91e3iT2 1 + (-17.9 - 17.9i)T + 4.91e3iT^{2}
19 1−149.T+6.85e3T2 1 - 149.T + 6.85e3T^{2}
23 1+(−44.6+44.6i)T−1.21e4iT2 1 + (-44.6 + 44.6i)T - 1.21e4iT^{2}
29 1+145.iT−2.43e4T2 1 + 145. iT - 2.43e4T^{2}
31 1−102.T+2.97e4T2 1 - 102.T + 2.97e4T^{2}
37 1+(−2.90−2.90i)T+5.06e4iT2 1 + (-2.90 - 2.90i)T + 5.06e4iT^{2}
41 1+163.iT−6.89e4T2 1 + 163. iT - 6.89e4T^{2}
43 1+(−251.+251.i)T−7.95e4iT2 1 + (-251. + 251. i)T - 7.95e4iT^{2}
47 1+(−269.−269.i)T+1.03e5iT2 1 + (-269. - 269. i)T + 1.03e5iT^{2}
53 1+(−411.−411.i)T+1.48e5iT2 1 + (-411. - 411. i)T + 1.48e5iT^{2}
59 1+91.0iT−2.05e5T2 1 + 91.0iT - 2.05e5T^{2}
61 1−602.iT−2.26e5T2 1 - 602. iT - 2.26e5T^{2}
67 1+(571.+571.i)T+3.00e5iT2 1 + (571. + 571. i)T + 3.00e5iT^{2}
71 1−802.iT−3.57e5T2 1 - 802. iT - 3.57e5T^{2}
73 1+(−348.−348.i)T+3.89e5iT2 1 + (-348. - 348. i)T + 3.89e5iT^{2}
79 1+923.iT−4.93e5T2 1 + 923. iT - 4.93e5T^{2}
83 1+(501.+501.i)T+5.71e5iT2 1 + (501. + 501. i)T + 5.71e5iT^{2}
89 1+178.T+7.04e5T2 1 + 178.T + 7.04e5T^{2}
97 1+(82.1−82.1i)T−9.12e5iT2 1 + (82.1 - 82.1i)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

−12.62056383603708346127722188842, −11.91833781574543455206229966855, −10.58537404779568957370263785012, −9.827834564367489941250252943530, −8.827125660357668945137898236467, −7.45129559398175102098756576485, −6.03778993174659842786933682298, −5.58637129313186704313401665445, −2.65576780959218174488410195866, −0.885189319556432306781799910127, 1.04296968292396136023994196959, 3.28415816451694882838693650965, 5.49034475729711459923900213021, 6.59763545416809245685090915059, 7.32921804724710367404631299494, 9.318156618231400206499904181299, 9.959194569942108880227522698009, 10.71703284602521542850939067199, 11.63771492675840689153463538889, 12.99713693109497052953720667639

Graph of the ZZ-function along the critical line