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Properties

Label 2-120-120.77-c3-0-2
Degree 22
Conductor 120120
Sign −0.968+0.248i-0.968 + 0.248i
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 − 2i)2-s + (−3.67 + 3.67i)3-s + 8i·4-s + (−3.32 + 10.6i)5-s + 14.6·6-s + (9.65 + 9.65i)7-s + (16 − 16i)8-s − 27i·9-s + (28 − 14.6i)10-s − 47.4·11-s + (−29.3 − 29.3i)12-s − 38.6i·14-s + (−26.9 − 51.4i)15-s − 64·16-s + (−54 + 54i)18-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.297 + 0.954i)5-s + 0.999·6-s + (0.521 + 0.521i)7-s + (0.707 − 0.707i)8-s − i·9-s + (0.885 − 0.464i)10-s − 1.30·11-s + (−0.707 − 0.707i)12-s − 0.736i·14-s + (−0.464 − 0.885i)15-s − 16-s + (−0.707 + 0.707i)18-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((−0.968+0.248i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=((−0.968+0.248i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −0.968+0.248i-0.968 + 0.248i
Analytic conductor: 7.080227.08022
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ120(77,⋅)\chi_{120} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :3/2), −0.968+0.248i)(2,\ 120,\ (\ :3/2),\ -0.968 + 0.248i)

Particular Values

L(2)L(2) ≈\approx 0.0130544−0.103249i0.0130544 - 0.103249i
L(12)L(\frac12) ≈\approx 0.0130544−0.103249i0.0130544 - 0.103249i
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+2i)T 1 + (2 + 2i)T
3 1+(3.67−3.67i)T 1 + (3.67 - 3.67i)T
5 1+(3.32−10.6i)T 1 + (3.32 - 10.6i)T
good7 1+(−9.65−9.65i)T+343iT2 1 + (-9.65 - 9.65i)T + 343iT^{2}
11 1+47.4T+1.33e3T2 1 + 47.4T + 1.33e3T^{2}
13 1+2.19e3iT2 1 + 2.19e3iT^{2}
17 1−4.91e3iT2 1 - 4.91e3iT^{2}
19 1+6.85e3T2 1 + 6.85e3T^{2}
23 1+1.21e4iT2 1 + 1.21e4iT^{2}
29 1+312.iT−2.43e4T2 1 + 312. iT - 2.43e4T^{2}
31 1+338.T+2.97e4T2 1 + 338.T + 2.97e4T^{2}
37 1−5.06e4iT2 1 - 5.06e4iT^{2}
41 1−6.89e4T2 1 - 6.89e4T^{2}
43 1+7.95e4iT2 1 + 7.95e4iT^{2}
47 1−1.03e5iT2 1 - 1.03e5iT^{2}
53 1+(−360.+360.i)T−1.48e5iT2 1 + (-360. + 360. i)T - 1.48e5iT^{2}
59 1−899.iT−2.05e5T2 1 - 899. iT - 2.05e5T^{2}
61 1−2.26e5T2 1 - 2.26e5T^{2}
67 1−3.00e5iT2 1 - 3.00e5iT^{2}
71 1−3.57e5T2 1 - 3.57e5T^{2}
73 1+(763.−763.i)T−3.89e5iT2 1 + (763. - 763. i)T - 3.89e5iT^{2}
79 1−308.iT−4.93e5T2 1 - 308. iT - 4.93e5T^{2}
83 1+(868−868i)T−5.71e5iT2 1 + (868 - 868i)T - 5.71e5iT^{2}
89 1+7.04e5T2 1 + 7.04e5T^{2}
97 1+(−1.19e3−1.19e3i)T+9.12e5iT2 1 + (-1.19e3 - 1.19e3i)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

−13.27015952761502944922078242428, −12.02205241839574485141460700410, −11.27396639348325094990504497311, −10.54873204946661654963510827758, −9.705719910608478520077198908861, −8.339898851765800389980323611132, −7.20607596472202746268658492992, −5.58504697290502058351313414914, −4.00212190583182385064155963610, −2.52883773183139141570356207795, 0.07587764675362501251942622142, 1.54433763187092646403498301697, 4.84605251233577030099586435630, 5.60237259640998413285340567490, 7.21011272407411837351803454659, 7.86301827116210954863857899850, 8.922619634932462451964436626998, 10.46903385830415368907839869549, 11.17485898637323665034612810432, 12.55522510246796760262041183975

Graph of the ZZ-function along the critical line