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Properties

Label 2-2160-15.14-c2-0-35
Degree 22
Conductor 21602160
Sign −0.882−0.469i-0.882 - 0.469i
Analytic cond. 58.855758.8557
Root an. cond. 7.671747.67174
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.41 + 2.34i)5-s + 13.6i·7-s + 12.3i·11-s + 17.0i·13-s + 6.89·17-s + 7.24·19-s − 34.7·23-s + (13.9 + 20.7i)25-s + 21.1i·29-s + 38.2·31-s + (−32.1 + 60.4i)35-s + 21.5i·37-s − 36.3i·41-s + 6.23i·43-s + 40.2·47-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)5-s + 1.95i·7-s + 1.11i·11-s + 1.30i·13-s + 0.405·17-s + 0.381·19-s − 1.51·23-s + (0.558 + 0.829i)25-s + 0.728i·29-s + 1.23·31-s + (−0.918 + 1.72i)35-s + 0.582i·37-s − 0.886i·41-s + 0.145i·43-s + 0.855·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.882−0.469i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((−0.882−0.469i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.882−0.469i-0.882 - 0.469i
Analytic conductor: 58.855758.8557
Root analytic conductor: 7.671747.67174
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :1), −0.882−0.469i)(2,\ 2160,\ (\ :1),\ -0.882 - 0.469i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.2937753602.293775360
L(12)L(\frac12) ≈\approx 2.2937753602.293775360
L(2)L(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
3 1 1
5 1+(−4.41−2.34i)T 1 + (-4.41 - 2.34i)T
good7 1−13.6iT−49T2 1 - 13.6iT - 49T^{2}
11 1−12.3iT−121T2 1 - 12.3iT - 121T^{2}
13 1−17.0iT−169T2 1 - 17.0iT - 169T^{2}
17 1−6.89T+289T2 1 - 6.89T + 289T^{2}
19 1−7.24T+361T2 1 - 7.24T + 361T^{2}
23 1+34.7T+529T2 1 + 34.7T + 529T^{2}
29 1−21.1iT−841T2 1 - 21.1iT - 841T^{2}
31 1−38.2T+961T2 1 - 38.2T + 961T^{2}
37 1−21.5iT−1.36e3T2 1 - 21.5iT - 1.36e3T^{2}
41 1+36.3iT−1.68e3T2 1 + 36.3iT - 1.68e3T^{2}
43 1−6.23iT−1.84e3T2 1 - 6.23iT - 1.84e3T^{2}
47 1−40.2T+2.20e3T2 1 - 40.2T + 2.20e3T^{2}
53 1−38.2T+2.80e3T2 1 - 38.2T + 2.80e3T^{2}
59 1+41.6iT−3.48e3T2 1 + 41.6iT - 3.48e3T^{2}
61 1+15.0T+3.72e3T2 1 + 15.0T + 3.72e3T^{2}
67 1+128.iT−4.48e3T2 1 + 128. iT - 4.48e3T^{2}
71 1+104.iT−5.04e3T2 1 + 104. iT - 5.04e3T^{2}
73 1+2.11iT−5.32e3T2 1 + 2.11iT - 5.32e3T^{2}
79 1−44.0T+6.24e3T2 1 - 44.0T + 6.24e3T^{2}
83 1−55.0T+6.88e3T2 1 - 55.0T + 6.88e3T^{2}
89 1+68.1iT−7.92e3T2 1 + 68.1iT - 7.92e3T^{2}
97 1+101.iT−9.40e3T2 1 + 101. iT - 9.40e3T^{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

−9.341688212872702453552199754202, −8.644125814061792401887423500300, −7.67562160207044368056407520325, −6.64603555963723480953343314814, −6.13774839589161563946079705719, −5.34042217646076370073087765278, −4.57551437668375885572538932097, −3.21804391508206378893349653582, −2.16910751017547926740051878067, −1.83984433107337903705633117549, 0.60640116098853111664703144823, 1.12527015810996137296118657494, 2.65947114816755415314769083109, 3.68859255214530588927471863901, 4.42531783591077676190979215805, 5.54322390408332456167204892481, 6.06246853741933668715394170756, 7.03867285917107634325300166543, 7.962572506284074932328621006035, 8.314417041526402608140150653485

Graph of the ZZ-function along the critical line