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Properties

Label 2-2160-12.11-c3-0-11
Degree 22
Conductor 21602160
Sign −1-1
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: −1-1
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 0.89011270410.8901127041
L(12)L(\frac12) ≈\approx 0.89011270410.8901127041
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

−9.148701862920846544285939254152, −8.496245740881756200056824224534, −7.40629457137832818037287898564, −6.82074837197660996824644662399, −5.99791061500127688941836083757, −5.15295650009407186980454866880, −4.35056165086576368470179048200, −3.10462251469429307071810747416, −2.50049631462422821294937239586, −1.33735204124623489846335512415, 0.18512213008751159904039247649, 1.17473480166577947928305689713, 2.19690992331681416789362765604, 3.63908584709686883285952056317, 4.13223003103368139136883782474, 5.05968999715628395222260700937, 5.97381611044847364163438242546, 6.95800716291271090520981687587, 7.43107958000274690565680051857, 8.349763197766835549413694720404

Graph of the ZZ-function along the critical line