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Properties

Label 2-120-15.2-c3-0-14
Degree 22
Conductor 120120
Sign −0.514+0.857i-0.514 + 0.857i
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  + (1.89 + 4.83i)3-s + (−8.68 − 7.04i)5-s + (−20.0 − 20.0i)7-s + (−19.8 + 18.3i)9-s − 46.6i·11-s + (−20.2 + 20.2i)13-s + (17.6 − 55.3i)15-s + (59.4 − 59.4i)17-s + 81.9i·19-s + (58.8 − 134. i)21-s + (−98.2 − 98.2i)23-s + (25.8 + 122. i)25-s + (−126. − 61.1i)27-s − 18.6·29-s − 278.·31-s + ⋯
L(s)  = 1  + (0.364 + 0.931i)3-s + (−0.776 − 0.629i)5-s + (−1.07 − 1.07i)7-s + (−0.733 + 0.679i)9-s − 1.27i·11-s + (−0.433 + 0.433i)13-s + (0.303 − 0.952i)15-s + (0.848 − 0.848i)17-s + 0.989i·19-s + (0.611 − 1.39i)21-s + (−0.890 − 0.890i)23-s + (0.206 + 0.978i)25-s + (−0.900 − 0.435i)27-s − 0.119·29-s − 1.61·31-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((−0.514+0.857i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((−0.514+0.857i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −0.514+0.857i-0.514 + 0.857i
Analytic conductor: 7.080227.08022
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ120(17,⋅)\chi_{120} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :3/2), −0.514+0.857i)(2,\ 120,\ (\ :3/2),\ -0.514 + 0.857i)

Particular Values

L(2)L(2) ≈\approx 0.279148−0.492711i0.279148 - 0.492711i
L(12)L(\frac12) ≈\approx 0.279148−0.492711i0.279148 - 0.492711i
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 1
3 1+(−1.89−4.83i)T 1 + (-1.89 - 4.83i)T
5 1+(8.68+7.04i)T 1 + (8.68 + 7.04i)T
good7 1+(20.0+20.0i)T+343iT2 1 + (20.0 + 20.0i)T + 343iT^{2}
11 1+46.6iT−1.33e3T2 1 + 46.6iT - 1.33e3T^{2}
13 1+(20.2−20.2i)T−2.19e3iT2 1 + (20.2 - 20.2i)T - 2.19e3iT^{2}
17 1+(−59.4+59.4i)T−4.91e3iT2 1 + (-59.4 + 59.4i)T - 4.91e3iT^{2}
19 1−81.9iT−6.85e3T2 1 - 81.9iT - 6.85e3T^{2}
23 1+(98.2+98.2i)T+1.21e4iT2 1 + (98.2 + 98.2i)T + 1.21e4iT^{2}
29 1+18.6T+2.43e4T2 1 + 18.6T + 2.43e4T^{2}
31 1+278.T+2.97e4T2 1 + 278.T + 2.97e4T^{2}
37 1+(81.9+81.9i)T+5.06e4iT2 1 + (81.9 + 81.9i)T + 5.06e4iT^{2}
41 1−211.iT−6.89e4T2 1 - 211. iT - 6.89e4T^{2}
43 1+(−168.+168.i)T−7.95e4iT2 1 + (-168. + 168. i)T - 7.95e4iT^{2}
47 1+(−24.9+24.9i)T−1.03e5iT2 1 + (-24.9 + 24.9i)T - 1.03e5iT^{2}
53 1+(−54.7−54.7i)T+1.48e5iT2 1 + (-54.7 - 54.7i)T + 1.48e5iT^{2}
59 1−158.T+2.05e5T2 1 - 158.T + 2.05e5T^{2}
61 1−892.T+2.26e5T2 1 - 892.T + 2.26e5T^{2}
67 1+(407.+407.i)T+3.00e5iT2 1 + (407. + 407. i)T + 3.00e5iT^{2}
71 1+286.iT−3.57e5T2 1 + 286. iT - 3.57e5T^{2}
73 1+(588.−588.i)T−3.89e5iT2 1 + (588. - 588. i)T - 3.89e5iT^{2}
79 1−693.iT−4.93e5T2 1 - 693. iT - 4.93e5T^{2}
83 1+(735.+735.i)T+5.71e5iT2 1 + (735. + 735. i)T + 5.71e5iT^{2}
89 1−755.T+7.04e5T2 1 - 755.T + 7.04e5T^{2}
97 1+(760.+760.i)T+9.12e5iT2 1 + (760. + 760. i)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

−12.68661014565850994704078945593, −11.51119443901289550227031774918, −10.42364970445023077209799525371, −9.528191057762947634107358335444, −8.451111268655364727766346337073, −7.34545932311140986409975478098, −5.64080693879029081583454343302, −4.12078189837982770078230481901, −3.34617403768294319892882250485, −0.27242399169493834737290162625, 2.31501779468854748290036924502, 3.53544682960389209331405551847, 5.71833837588931063526625267584, 6.93616187571430855995880258631, 7.71899680299683071876448862046, 9.018311643140293079122654197034, 10.08788023167173082771784977776, 11.66125122980018820718647652284, 12.42445194314424075055329020897, 13.00852160971514639590434839113

Graph of the ZZ-function along the critical line