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Properties

Label 2-120-40.29-c1-0-9
Degree 22
Conductor 120120
Sign 0.756+0.654i0.756 + 0.654i
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  + (1.34 − 0.450i)2-s − 3-s + (1.59 − 1.20i)4-s + (0.254 − 2.22i)5-s + (−1.34 + 0.450i)6-s + 2.64i·7-s + (1.59 − 2.33i)8-s + 9-s + (−0.659 − 3.09i)10-s + 1.51i·11-s + (−1.59 + 1.20i)12-s − 3.87·13-s + (1.18 + 3.54i)14-s + (−0.254 + 2.22i)15-s + (1.08 − 3.84i)16-s + 3.31i·17-s + ⋯
L(s)  = 1  + (0.947 − 0.318i)2-s − 0.577·3-s + (0.797 − 0.603i)4-s + (0.113 − 0.993i)5-s + (−0.547 + 0.183i)6-s + 0.998i·7-s + (0.563 − 0.825i)8-s + 0.333·9-s + (−0.208 − 0.978i)10-s + 0.456i·11-s + (−0.460 + 0.348i)12-s − 1.07·13-s + (0.317 + 0.946i)14-s + (−0.0656 + 0.573i)15-s + (0.271 − 0.962i)16-s + 0.803i·17-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=((0.756+0.654i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+1/2) L(s)=((0.756+0.654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 0.756+0.654i0.756 + 0.654i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ120(109,⋅)\chi_{120} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 120, ( :1/2), 0.756+0.654i)(2,\ 120,\ (\ :1/2),\ 0.756 + 0.654i)

Particular Values

L(1)L(1) ≈\approx 1.40525−0.523259i1.40525 - 0.523259i
L(12)L(\frac12) ≈\approx 1.40525−0.523259i1.40525 - 0.523259i
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+(−1.34+0.450i)T 1 + (-1.34 + 0.450i)T
3 1+T 1 + T
5 1+(−0.254+2.22i)T 1 + (-0.254 + 2.22i)T
good7 1−2.64iT−7T2 1 - 2.64iT - 7T^{2}
11 1−1.51iT−11T2 1 - 1.51iT - 11T^{2}
13 1+3.87T+13T2 1 + 3.87T + 13T^{2}
17 1−3.31iT−17T2 1 - 3.31iT - 17T^{2}
19 1−7.08iT−19T2 1 - 7.08iT - 19T^{2}
23 1+4.82iT−23T2 1 + 4.82iT - 23T^{2}
29 1+2.18iT−29T2 1 + 2.18iT - 29T^{2}
31 1+7.36T+31T2 1 + 7.36T + 31T^{2}
37 1−7.87T+37T2 1 - 7.87T + 37T^{2}
41 1−8.72T+41T2 1 - 8.72T + 41T^{2}
43 1+1.01T+43T2 1 + 1.01T + 43T^{2}
47 1+7.08iT−47T2 1 + 7.08iT - 47T^{2}
53 1+4.50T+53T2 1 + 4.50T + 53T^{2}
59 1+6.79iT−59T2 1 + 6.79iT - 59T^{2}
61 1−3.60iT−61T2 1 - 3.60iT - 61T^{2}
67 1−1.01T+67T2 1 - 1.01T + 67T^{2}
71 1+6.72T+71T2 1 + 6.72T + 71T^{2}
73 1+15.5iT−73T2 1 + 15.5iT - 73T^{2}
79 1−7.36T+79T2 1 - 7.36T + 79T^{2}
83 1+7.74T+83T2 1 + 7.74T + 83T^{2}
89 1+14.7T+89T2 1 + 14.7T + 89T^{2}
97 1−11.1iT−97T2 1 - 11.1iT - 97T^{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

−12.85215749268182238244952741735, −12.47746165405455465672646929024, −11.76372014877420549778640051255, −10.38676535116238017890945237812, −9.387682569787316647057491080489, −7.81777905856824308553279539793, −6.15148969379611355540884927692, −5.33880860225500696184261446878, −4.22098229968495723923824090517, −2.04128664062871564010657097566, 2.87615994388096863389644470442, 4.39508971616457155186481528911, 5.69149156033938695983616953089, 7.05349848888652327567376920367, 7.41723458285377844873052520256, 9.637646177083581400572635359735, 11.02618866982096631940481939745, 11.32529500911265445738641507066, 12.76264528448803874740282446109, 13.70209765636108661839520119753

Graph of the ZZ-function along the critical line