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Properties

Label 2-120-120.53-c3-0-40
Degree 22
Conductor 120120
Sign 0.722+0.691i0.722 + 0.691i
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  + (−2.22 + 1.74i)2-s + (1.34 − 5.01i)3-s + (1.88 − 7.77i)4-s + (10.3 + 4.10i)5-s + (5.78 + 13.5i)6-s + (2.75 − 2.75i)7-s + (9.39 + 20.5i)8-s + (−23.3 − 13.5i)9-s + (−30.2 + 9.05i)10-s + 18.4·11-s + (−36.4 − 19.9i)12-s + (22.4 − 22.4i)13-s + (−1.31 + 10.9i)14-s + (34.5 − 46.6i)15-s + (−56.8 − 29.3i)16-s + (−73.3 − 73.3i)17-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (0.258 − 0.965i)3-s + (0.236 − 0.971i)4-s + (0.930 + 0.367i)5-s + (0.393 + 0.919i)6-s + (0.148 − 0.148i)7-s + (0.415 + 0.909i)8-s + (−0.865 − 0.500i)9-s + (−0.958 + 0.286i)10-s + 0.504·11-s + (−0.877 − 0.479i)12-s + (0.478 − 0.478i)13-s + (−0.0250 + 0.208i)14-s + (0.595 − 0.803i)15-s + (−0.888 − 0.458i)16-s + (−1.04 − 1.04i)17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.722+0.691i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((0.722+0.691i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.722+0.691i0.722 + 0.691i
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), 0.722+0.691i)(2,\ 120,\ (\ :3/2),\ 0.722 + 0.691i)

Particular Values

L(2)L(2) ≈\approx 1.28613−0.516296i1.28613 - 0.516296i
L(12)L(\frac12) ≈\approx 1.28613−0.516296i1.28613 - 0.516296i
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+(2.22−1.74i)T 1 + (2.22 - 1.74i)T
3 1+(−1.34+5.01i)T 1 + (-1.34 + 5.01i)T
5 1+(−10.3−4.10i)T 1 + (-10.3 - 4.10i)T
good7 1+(−2.75+2.75i)T−343iT2 1 + (-2.75 + 2.75i)T - 343iT^{2}
11 1−18.4T+1.33e3T2 1 - 18.4T + 1.33e3T^{2}
13 1+(−22.4+22.4i)T−2.19e3iT2 1 + (-22.4 + 22.4i)T - 2.19e3iT^{2}
17 1+(73.3+73.3i)T+4.91e3iT2 1 + (73.3 + 73.3i)T + 4.91e3iT^{2}
19 1−86.5T+6.85e3T2 1 - 86.5T + 6.85e3T^{2}
23 1+(−102.+102.i)T−1.21e4iT2 1 + (-102. + 102. i)T - 1.21e4iT^{2}
29 1+28.7iT−2.43e4T2 1 + 28.7iT - 2.43e4T^{2}
31 1−140.T+2.97e4T2 1 - 140.T + 2.97e4T^{2}
37 1+(242.+242.i)T+5.06e4iT2 1 + (242. + 242. i)T + 5.06e4iT^{2}
41 1+52.3iT−6.89e4T2 1 + 52.3iT - 6.89e4T^{2}
43 1+(16.8−16.8i)T−7.95e4iT2 1 + (16.8 - 16.8i)T - 7.95e4iT^{2}
47 1+(298.+298.i)T+1.03e5iT2 1 + (298. + 298. i)T + 1.03e5iT^{2}
53 1+(−49.3−49.3i)T+1.48e5iT2 1 + (-49.3 - 49.3i)T + 1.48e5iT^{2}
59 1−361.iT−2.05e5T2 1 - 361. iT - 2.05e5T^{2}
61 1−437.iT−2.26e5T2 1 - 437. iT - 2.26e5T^{2}
67 1+(−754.−754.i)T+3.00e5iT2 1 + (-754. - 754. i)T + 3.00e5iT^{2}
71 1−554.iT−3.57e5T2 1 - 554. iT - 3.57e5T^{2}
73 1+(564.+564.i)T+3.89e5iT2 1 + (564. + 564. i)T + 3.89e5iT^{2}
79 1−621.iT−4.93e5T2 1 - 621. iT - 4.93e5T^{2}
83 1+(−569.−569.i)T+5.71e5iT2 1 + (-569. - 569. i)T + 5.71e5iT^{2}
89 1−412.T+7.04e5T2 1 - 412.T + 7.04e5T^{2}
97 1+(653.−653.i)T−9.12e5iT2 1 + (653. - 653. 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

−13.24787852608496276434837831736, −11.67822585670299031660426896155, −10.65810098434961579169637984615, −9.383999513095686745088965669021, −8.596740202453354593699811365274, −7.22619107384198812247499798490, −6.55537630254491392519749826086, −5.36586092517497772779553086927, −2.56022459497695182852188801658, −1.02228063156381216524218753183, 1.70227309349962013048501270453, 3.36841834654090656333125709920, 4.84480404106722713642841456167, 6.48928236069212048964770682377, 8.316126963835814612901255019645, 9.110368249834811458660029474468, 9.815360187264139890074832235742, 10.86391137435729807796113849974, 11.73378687776385411516384690854, 13.14323779089113085124926557646

Graph of the ZZ-function along the critical line