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Properties

Label 2-2160-15.14-c2-0-56
Degree 22
Conductor 21602160
Sign 0.572+0.819i0.572 + 0.819i
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  + (−2.86 − 4.09i)5-s − 7.15i·7-s + 5.06i·11-s + 3.12i·13-s + 8.72·17-s + 20.1·19-s + 14.7·23-s + (−8.59 + 23.4i)25-s + 39.7i·29-s + 39.3·31-s + (−29.3 + 20.4i)35-s + 34.8i·37-s + 13.2i·41-s − 66.6i·43-s + 16.9·47-s + ⋯
L(s)  = 1  + (−0.572 − 0.819i)5-s − 1.02i·7-s + 0.460i·11-s + 0.240i·13-s + 0.513·17-s + 1.06·19-s + 0.640·23-s + (−0.343 + 0.939i)25-s + 1.37i·29-s + 1.27·31-s + (−0.837 + 0.585i)35-s + 0.942i·37-s + 0.323i·41-s − 1.54i·43-s + 0.359·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.572+0.819i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+1) L(s)=((0.572+0.819i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.572+0.819i0.572 + 0.819i
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), 0.572+0.819i)(2,\ 2160,\ (\ :1),\ 0.572 + 0.819i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.8823983871.882398387
L(12)L(\frac12) ≈\approx 1.8823983871.882398387
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+(2.86+4.09i)T 1 + (2.86 + 4.09i)T
good7 1+7.15iT−49T2 1 + 7.15iT - 49T^{2}
11 1−5.06iT−121T2 1 - 5.06iT - 121T^{2}
13 1−3.12iT−169T2 1 - 3.12iT - 169T^{2}
17 1−8.72T+289T2 1 - 8.72T + 289T^{2}
19 1−20.1T+361T2 1 - 20.1T + 361T^{2}
23 1−14.7T+529T2 1 - 14.7T + 529T^{2}
29 1−39.7iT−841T2 1 - 39.7iT - 841T^{2}
31 1−39.3T+961T2 1 - 39.3T + 961T^{2}
37 1−34.8iT−1.36e3T2 1 - 34.8iT - 1.36e3T^{2}
41 1−13.2iT−1.68e3T2 1 - 13.2iT - 1.68e3T^{2}
43 1+66.6iT−1.84e3T2 1 + 66.6iT - 1.84e3T^{2}
47 1−16.9T+2.20e3T2 1 - 16.9T + 2.20e3T^{2}
53 1+4.62T+2.80e3T2 1 + 4.62T + 2.80e3T^{2}
59 1+25.7iT−3.48e3T2 1 + 25.7iT - 3.48e3T^{2}
61 1+12.1T+3.72e3T2 1 + 12.1T + 3.72e3T^{2}
67 1+106.iT−4.48e3T2 1 + 106. iT - 4.48e3T^{2}
71 1−101.iT−5.04e3T2 1 - 101. iT - 5.04e3T^{2}
73 1−23.2iT−5.32e3T2 1 - 23.2iT - 5.32e3T^{2}
79 1−66.5T+6.24e3T2 1 - 66.5T + 6.24e3T^{2}
83 1−144.T+6.88e3T2 1 - 144.T + 6.88e3T^{2}
89 1+154.iT−7.92e3T2 1 + 154. iT - 7.92e3T^{2}
97 1+175.iT−9.40e3T2 1 + 175. 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

−8.759552643896691667716435294393, −7.921026113320900113490307029415, −7.31247499164737837329738701604, −6.64292660081511594897101542535, −5.33266981132248607014922667984, −4.77196822507263347157883499006, −3.90534232403851293415952699847, −3.09713444608772066687689511379, −1.48737997847937688239799373727, −0.68546635599078927958049217665, 0.835810586421629522373645965072, 2.45767643306983300593775718562, 3.04831472959587823557731024274, 3.99880228810112708617311903516, 5.12956633168430795171826042771, 5.92154072264997632557879587901, 6.59284931437013747931149319132, 7.66678796860162947366493484387, 8.043237254524327591711167547383, 9.041331233647584744430062978265

Graph of the ZZ-function along the critical line