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Properties

Label 2-120-120.59-c3-0-6
Degree 22
Conductor 120120
Sign 0.574−0.818i0.574 - 0.818i
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  + (−2.73 − 0.732i)2-s + (−2.64 − 4.47i)3-s + (6.92 + 4.00i)4-s + (−10.4 − 3.92i)5-s + (3.93 + 14.1i)6-s − 4.10·7-s + (−15.9 − 16.0i)8-s + (−13.0 + 23.6i)9-s + (25.7 + 18.4i)10-s − 5.96i·11-s + (−0.378 − 41.5i)12-s − 33.0·13-s + (11.2 + 3.00i)14-s + (10.0 + 57.2i)15-s + (31.9 + 55.4i)16-s + 50.6·17-s + ⋯
L(s)  = 1  + (−0.965 − 0.259i)2-s + (−0.508 − 0.861i)3-s + (0.865 + 0.500i)4-s + (−0.936 − 0.351i)5-s + (0.267 + 0.963i)6-s − 0.221·7-s + (−0.706 − 0.707i)8-s + (−0.483 + 0.875i)9-s + (0.813 + 0.582i)10-s − 0.163i·11-s + (−0.00910 − 0.999i)12-s − 0.705·13-s + (0.213 + 0.0573i)14-s + (0.173 + 0.984i)15-s + (0.498 + 0.866i)16-s + 0.722·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.574−0.818i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.574−0.818i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.574−0.818i0.574 - 0.818i
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), 0.574−0.818i)(2,\ 120,\ (\ :3/2),\ 0.574 - 0.818i)

Particular Values

L(2)L(2) ≈\approx 0.292349+0.151956i0.292349 + 0.151956i
L(12)L(\frac12) ≈\approx 0.292349+0.151956i0.292349 + 0.151956i
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+(2.73+0.732i)T 1 + (2.73 + 0.732i)T
3 1+(2.64+4.47i)T 1 + (2.64 + 4.47i)T
5 1+(10.4+3.92i)T 1 + (10.4 + 3.92i)T
good7 1+4.10T+343T2 1 + 4.10T + 343T^{2}
11 1+5.96iT−1.33e3T2 1 + 5.96iT - 1.33e3T^{2}
13 1+33.0T+2.19e3T2 1 + 33.0T + 2.19e3T^{2}
17 1−50.6T+4.91e3T2 1 - 50.6T + 4.91e3T^{2}
19 1−74.1T+6.85e3T2 1 - 74.1T + 6.85e3T^{2}
23 1−184.iT−1.21e4T2 1 - 184. iT - 1.21e4T^{2}
29 1−98.3T+2.43e4T2 1 - 98.3T + 2.43e4T^{2}
31 1−192.iT−2.97e4T2 1 - 192. iT - 2.97e4T^{2}
37 1+350.T+5.06e4T2 1 + 350.T + 5.06e4T^{2}
41 1+292.iT−6.89e4T2 1 + 292. iT - 6.89e4T^{2}
43 1−80.8iT−7.95e4T2 1 - 80.8iT - 7.95e4T^{2}
47 1−63.1iT−1.03e5T2 1 - 63.1iT - 1.03e5T^{2}
53 1−178.iT−1.48e5T2 1 - 178. iT - 1.48e5T^{2}
59 1−479.iT−2.05e5T2 1 - 479. iT - 2.05e5T^{2}
61 1+635.iT−2.26e5T2 1 + 635. iT - 2.26e5T^{2}
67 1−288.iT−3.00e5T2 1 - 288. iT - 3.00e5T^{2}
71 1+1.06e3T+3.57e5T2 1 + 1.06e3T + 3.57e5T^{2}
73 1−980.iT−3.89e5T2 1 - 980. iT - 3.89e5T^{2}
79 1−804.iT−4.93e5T2 1 - 804. iT - 4.93e5T^{2}
83 1+300.T+5.71e5T2 1 + 300.T + 5.71e5T^{2}
89 1−1.11e3iT−7.04e5T2 1 - 1.11e3iT - 7.04e5T^{2}
97 1+1.21e3iT−9.12e5T2 1 + 1.21e3iT - 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

−12.62005549185717141557481848226, −12.01938987103332848961444003250, −11.27685811184156026106238034749, −10.05480296205942801498577626279, −8.729779680528889249618856081773, −7.65509632753382474383994738663, −7.03216159909661847764815440724, −5.39610415419522465872608032524, −3.22757870017109516258418884460, −1.24371727221496961552898729459, 0.28375590288873179088977729356, 3.10764554988738305071219371476, 4.82200016973143083633945276065, 6.31313875771962052840046272994, 7.45090198508506542650703015435, 8.619209562147149906820876030682, 9.843632974461782980070688201641, 10.51299509462586514356354249960, 11.62524641720604796176334539118, 12.25352583619018570446078950331

Graph of the ZZ-function along the critical line