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Properties

Label 2-120-120.59-c3-0-10
Degree 22
Conductor 120120
Sign −0.916+0.399i-0.916 + 0.399i
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  + (−2.12 + 1.86i)2-s + (0.659 + 5.15i)3-s + (1.02 − 7.93i)4-s + (10.6 + 3.40i)5-s + (−11.0 − 9.71i)6-s − 28.2·7-s + (12.6 + 18.7i)8-s + (−26.1 + 6.79i)9-s + (−28.9 + 12.6i)10-s + 38.5i·11-s + (41.5 + 0.0568i)12-s + 36.1·13-s + (60.0 − 52.7i)14-s + (−10.5 + 57.1i)15-s + (−61.8 − 16.2i)16-s − 74.9·17-s + ⋯
L(s)  = 1  + (−0.751 + 0.660i)2-s + (0.126 + 0.991i)3-s + (0.128 − 0.991i)4-s + (0.952 + 0.304i)5-s + (−0.750 − 0.661i)6-s − 1.52·7-s + (0.558 + 0.829i)8-s + (−0.967 + 0.251i)9-s + (−0.916 + 0.400i)10-s + 1.05i·11-s + (0.999 + 0.00136i)12-s + 0.772·13-s + (1.14 − 1.00i)14-s + (−0.180 + 0.983i)15-s + (−0.967 − 0.254i)16-s − 1.06·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((−0.916+0.399i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((−0.916+0.399i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −0.916+0.399i-0.916 + 0.399i
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), −0.916+0.399i)(2,\ 120,\ (\ :3/2),\ -0.916 + 0.399i)

Particular Values

L(2)L(2) ≈\approx 0.131671−0.632380i0.131671 - 0.632380i
L(12)L(\frac12) ≈\approx 0.131671−0.632380i0.131671 - 0.632380i
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+(2.12−1.86i)T 1 + (2.12 - 1.86i)T
3 1+(−0.659−5.15i)T 1 + (-0.659 - 5.15i)T
5 1+(−10.6−3.40i)T 1 + (-10.6 - 3.40i)T
good7 1+28.2T+343T2 1 + 28.2T + 343T^{2}
11 1−38.5iT−1.33e3T2 1 - 38.5iT - 1.33e3T^{2}
13 1−36.1T+2.19e3T2 1 - 36.1T + 2.19e3T^{2}
17 1+74.9T+4.91e3T2 1 + 74.9T + 4.91e3T^{2}
19 1+136.T+6.85e3T2 1 + 136.T + 6.85e3T^{2}
23 1+114.iT−1.21e4T2 1 + 114. iT - 1.21e4T^{2}
29 1−109.T+2.43e4T2 1 - 109.T + 2.43e4T^{2}
31 1−57.1iT−2.97e4T2 1 - 57.1iT - 2.97e4T^{2}
37 1+100.T+5.06e4T2 1 + 100.T + 5.06e4T^{2}
41 1−173.iT−6.89e4T2 1 - 173. iT - 6.89e4T^{2}
43 1−86.0iT−7.95e4T2 1 - 86.0iT - 7.95e4T^{2}
47 1−239.iT−1.03e5T2 1 - 239. iT - 1.03e5T^{2}
53 1−476.iT−1.48e5T2 1 - 476. iT - 1.48e5T^{2}
59 1−762.iT−2.05e5T2 1 - 762. iT - 2.05e5T^{2}
61 1+614.iT−2.26e5T2 1 + 614. iT - 2.26e5T^{2}
67 1−382.iT−3.00e5T2 1 - 382. iT - 3.00e5T^{2}
71 1−124.T+3.57e5T2 1 - 124.T + 3.57e5T^{2}
73 1+267.iT−3.89e5T2 1 + 267. iT - 3.89e5T^{2}
79 1−586.iT−4.93e5T2 1 - 586. iT - 4.93e5T^{2}
83 1−282.T+5.71e5T2 1 - 282.T + 5.71e5T^{2}
89 1−604.iT−7.04e5T2 1 - 604. iT - 7.04e5T^{2}
97 1−1.04e3iT−9.12e5T2 1 - 1.04e3iT - 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

−13.81924295120473536523271192532, −12.77724708226569472123106738543, −10.78778758088421143630319141809, −10.24801366798614294425690673004, −9.368029222576211383134809561871, −8.645828354749741486238602958082, −6.67882061837249418587286764305, −6.13794946534348612098693464191, −4.50794006113644358474796734920, −2.49609643084857185970011568461, 0.39898368801665136970311037706, 2.09516696493404696789250174841, 3.42578455494788371976579617421, 6.10250121045089211593084364411, 6.76159578621633473742073865639, 8.513759897983760355415093670534, 9.033414444035140527576951093391, 10.27103295859436123264563734417, 11.32844545252836352191000080350, 12.64847753048013864472904258903

Graph of the ZZ-function along the critical line