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Properties

Label 2-2160-12.11-c3-0-36
Degree 22
Conductor 21602160
Sign 11
Analytic cond. 127.444127.444
Root an. cond. 11.289111.2891
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  + 5i·5-s − 20.7i·7-s − 36.3·11-s − 47·13-s − 21i·17-s + 62.3i·19-s − 36.3·23-s − 25·25-s + 123i·29-s − 25.9i·31-s + 103.·35-s − 178·37-s + 342i·41-s − 233. i·43-s − 306.·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.12i·7-s − 0.996·11-s − 1.00·13-s − 0.299i·17-s + 0.752i·19-s − 0.329·23-s − 0.200·25-s + 0.787i·29-s − 0.150i·31-s + 0.501·35-s − 0.790·37-s + 1.30i·41-s − 0.829i·43-s − 0.951·47-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=( Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s+3/2) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 11
Analytic conductor: 127.444127.444
Root analytic conductor: 11.289111.2891
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ2160(431,⋅)\chi_{2160} (431, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :3/2), 1)(2,\ 2160,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.3392176501.339217650
L(12)L(\frac12) ≈\approx 1.3392176501.339217650
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 1
3 1 1
5 1−5iT 1 - 5iT
good7 1+20.7iT−343T2 1 + 20.7iT - 343T^{2}
11 1+36.3T+1.33e3T2 1 + 36.3T + 1.33e3T^{2}
13 1+47T+2.19e3T2 1 + 47T + 2.19e3T^{2}
17 1+21iT−4.91e3T2 1 + 21iT - 4.91e3T^{2}
19 1−62.3iT−6.85e3T2 1 - 62.3iT - 6.85e3T^{2}
23 1+36.3T+1.21e4T2 1 + 36.3T + 1.21e4T^{2}
29 1−123iT−2.43e4T2 1 - 123iT - 2.43e4T^{2}
31 1+25.9iT−2.97e4T2 1 + 25.9iT - 2.97e4T^{2}
37 1+178T+5.06e4T2 1 + 178T + 5.06e4T^{2}
41 1−342iT−6.89e4T2 1 - 342iT - 6.89e4T^{2}
43 1+233.iT−7.95e4T2 1 + 233. iT - 7.95e4T^{2}
47 1+306.T+1.03e5T2 1 + 306.T + 1.03e5T^{2}
53 1−414iT−1.48e5T2 1 - 414iT - 1.48e5T^{2}
59 1−446.T+2.05e5T2 1 - 446.T + 2.05e5T^{2}
61 1−542T+2.26e5T2 1 - 542T + 2.26e5T^{2}
67 1+155.iT−3.00e5T2 1 + 155. iT - 3.00e5T^{2}
71 1−852.T+3.57e5T2 1 - 852.T + 3.57e5T^{2}
73 1−232T+3.89e5T2 1 - 232T + 3.89e5T^{2}
79 1+348.iT−4.93e5T2 1 + 348. iT - 4.93e5T^{2}
83 1−405.T+5.71e5T2 1 - 405.T + 5.71e5T^{2}
89 1+1.35e3iT−7.04e5T2 1 + 1.35e3iT - 7.04e5T^{2}
97 1+1.04e3T+9.12e5T2 1 + 1.04e3T + 9.12e5T^{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

−8.597922346956409181559980550539, −7.74484962312649799957413912575, −7.28941353585559926405029027930, −6.53965456238652224099596997339, −5.45844759429513828856444396760, −4.72945114364608256977946502882, −3.75987979241083916027447460738, −2.90289645067622904603312060405, −1.86302012185652922443439932310, −0.51967770900283116431401139961, 0.48513258045006321216966010445, 2.08315705161676714305022615568, 2.59389055792648094388307975367, 3.81583922044414399245772684419, 5.09259096925204621031731919691, 5.24081903078408761523283430502, 6.31469216724134642173155378116, 7.22355736709647078970420138073, 8.124927343483465054019370282503, 8.611557663943615536378818867728

Graph of the ZZ-function along the critical line