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Properties

Label 2-120-120.77-c1-0-13
Degree 22
Conductor 120120
Sign 0.980−0.195i0.980 - 0.195i
Analytic cond. 0.9582040.958204
Root an. cond. 0.9788790.978879
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 − 0.250i)2-s + (1.01 + 1.40i)3-s + (1.87 − 0.696i)4-s + (−2.23 + 0.116i)5-s + (1.76 + 1.69i)6-s + (−2.29 − 2.29i)7-s + (2.43 − 1.43i)8-s + (−0.925 + 2.85i)9-s + (−3.07 + 0.720i)10-s − 2.28·11-s + (2.88 + 1.91i)12-s + (1.05 + 1.05i)13-s + (−3.76 − 2.61i)14-s + (−2.43 − 3.00i)15-s + (3.03 − 2.61i)16-s + (3.04 − 3.04i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.176i)2-s + (0.588 + 0.808i)3-s + (0.937 − 0.348i)4-s + (−0.998 + 0.0519i)5-s + (0.721 + 0.692i)6-s + (−0.865 − 0.865i)7-s + (0.861 − 0.508i)8-s + (−0.308 + 0.951i)9-s + (−0.973 + 0.227i)10-s − 0.688·11-s + (0.832 + 0.553i)12-s + (0.292 + 0.292i)13-s + (−1.00 − 0.698i)14-s + (−0.629 − 0.777i)15-s + (0.757 − 0.652i)16-s + (0.738 − 0.738i)17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.980−0.195i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+1/2) L(s)=((0.980−0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.980−0.195i0.980 - 0.195i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ120(77,⋅)\chi_{120} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :1/2), 0.980−0.195i)(2,\ 120,\ (\ :1/2),\ 0.980 - 0.195i)

Particular Values

L(1)L(1) ≈\approx 1.73329+0.170982i1.73329 + 0.170982i
L(12)L(\frac12) ≈\approx 1.73329+0.170982i1.73329 + 0.170982i
L(32)L(\frac{3}{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+(−1.39+0.250i)T 1 + (-1.39 + 0.250i)T
3 1+(−1.01−1.40i)T 1 + (-1.01 - 1.40i)T
5 1+(2.23−0.116i)T 1 + (2.23 - 0.116i)T
good7 1+(2.29+2.29i)T+7iT2 1 + (2.29 + 2.29i)T + 7iT^{2}
11 1+2.28T+11T2 1 + 2.28T + 11T^{2}
13 1+(−1.05−1.05i)T+13iT2 1 + (-1.05 - 1.05i)T + 13iT^{2}
17 1+(−3.04+3.04i)T−17iT2 1 + (-3.04 + 3.04i)T - 17iT^{2}
19 1+3.36T+19T2 1 + 3.36T + 19T^{2}
23 1+(−3.68−3.68i)T+23iT2 1 + (-3.68 - 3.68i)T + 23iT^{2}
29 1−2.71iT−29T2 1 - 2.71iT - 29T^{2}
31 1+6.49T+31T2 1 + 6.49T + 31T^{2}
37 1+(−2.31+2.31i)T−37iT2 1 + (-2.31 + 2.31i)T - 37iT^{2}
41 1−10.8iT−41T2 1 - 10.8iT - 41T^{2}
43 1+(1.16+1.16i)T+43iT2 1 + (1.16 + 1.16i)T + 43iT^{2}
47 1+(1.83−1.83i)T−47iT2 1 + (1.83 - 1.83i)T - 47iT^{2}
53 1+(−5.82+5.82i)T−53iT2 1 + (-5.82 + 5.82i)T - 53iT^{2}
59 1+7.41iT−59T2 1 + 7.41iT - 59T^{2}
61 1−8.97iT−61T2 1 - 8.97iT - 61T^{2}
67 1+(−8.66+8.66i)T−67iT2 1 + (-8.66 + 8.66i)T - 67iT^{2}
71 1+7.37iT−71T2 1 + 7.37iT - 71T^{2}
73 1+(−1.83+1.83i)T−73iT2 1 + (-1.83 + 1.83i)T - 73iT^{2}
79 1−8.28iT−79T2 1 - 8.28iT - 79T^{2}
83 1+(5.27−5.27i)T−83iT2 1 + (5.27 - 5.27i)T - 83iT^{2}
89 1−11.5T+89T2 1 - 11.5T + 89T^{2}
97 1+(2.79+2.79i)T+97iT2 1 + (2.79 + 2.79i)T + 97iT^{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

−13.49952811692865906718761726837, −12.75230091495172076399976129348, −11.33953921043298250974351354973, −10.62398401980120016245846460735, −9.544880709057562103681074038640, −7.910558127086609728157796719407, −6.92617174673655988098345879685, −5.13278063036350406261352142616, −3.92096075349122477689781910497, −3.09713110015318820756830541090, 2.64982127043694289258862022966, 3.77338372295083905536486703812, 5.64412936629592742128139491280, 6.77583212803665820241976890697, 7.87092181815821649738201924799, 8.768900006681544833093619218259, 10.63643924654515772583486820642, 11.92567997588791401248898200788, 12.68611457031687562516319739614, 13.10833475658546002814765516394

Graph of the ZZ-function along the critical line