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Properties

Label 2-2160-1.1-c3-0-11
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 yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 24.7·7-s − 8.16·11-s − 46.7·13-s − 60.3·17-s + 111.·19-s + 36.9·23-s + 25·25-s − 33.2·29-s + 124.·31-s − 123.·35-s − 438.·37-s − 508.·41-s + 48.5·43-s − 248.·47-s + 271.·49-s + 320.·53-s − 40.8·55-s − 652.·59-s + 693.·61-s − 233.·65-s + 12.0·67-s + 1.16e3·71-s − 122.·73-s + 202.·77-s + 441.·79-s + 428.·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.33·7-s − 0.223·11-s − 0.997·13-s − 0.860·17-s + 1.34·19-s + 0.334·23-s + 0.200·25-s − 0.212·29-s + 0.720·31-s − 0.598·35-s − 1.94·37-s − 1.93·41-s + 0.172·43-s − 0.769·47-s + 0.791·49-s + 0.830·53-s − 0.100·55-s − 1.43·59-s + 1.45·61-s − 0.446·65-s + 0.0219·67-s + 1.95·71-s − 0.195·73-s + 0.299·77-s + 0.629·79-s + 0.566·83-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 & \, \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 & \, \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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2160, ( :3/2), 1)(2,\ 2160,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.2522280831.252228083
L(12)L(\frac12) ≈\approx 1.2522280831.252228083
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 1−5T 1 - 5T
good7 1+24.7T+343T2 1 + 24.7T + 343T^{2}
11 1+8.16T+1.33e3T2 1 + 8.16T + 1.33e3T^{2}
13 1+46.7T+2.19e3T2 1 + 46.7T + 2.19e3T^{2}
17 1+60.3T+4.91e3T2 1 + 60.3T + 4.91e3T^{2}
19 1−111.T+6.85e3T2 1 - 111.T + 6.85e3T^{2}
23 1−36.9T+1.21e4T2 1 - 36.9T + 1.21e4T^{2}
29 1+33.2T+2.43e4T2 1 + 33.2T + 2.43e4T^{2}
31 1−124.T+2.97e4T2 1 - 124.T + 2.97e4T^{2}
37 1+438.T+5.06e4T2 1 + 438.T + 5.06e4T^{2}
41 1+508.T+6.89e4T2 1 + 508.T + 6.89e4T^{2}
43 1−48.5T+7.95e4T2 1 - 48.5T + 7.95e4T^{2}
47 1+248.T+1.03e5T2 1 + 248.T + 1.03e5T^{2}
53 1−320.T+1.48e5T2 1 - 320.T + 1.48e5T^{2}
59 1+652.T+2.05e5T2 1 + 652.T + 2.05e5T^{2}
61 1−693.T+2.26e5T2 1 - 693.T + 2.26e5T^{2}
67 1−12.0T+3.00e5T2 1 - 12.0T + 3.00e5T^{2}
71 1−1.16e3T+3.57e5T2 1 - 1.16e3T + 3.57e5T^{2}
73 1+122.T+3.89e5T2 1 + 122.T + 3.89e5T^{2}
79 1−441.T+4.93e5T2 1 - 441.T + 4.93e5T^{2}
83 1−428.T+5.71e5T2 1 - 428.T + 5.71e5T^{2}
89 1−1.54e3T+7.04e5T2 1 - 1.54e3T + 7.04e5T^{2}
97 1+500.T+9.12e5T2 1 + 500.T + 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

−8.911657580914181102711911098388, −7.920816652842604096063382956568, −6.84567770276326696293066134168, −6.67121043329780711788790803197, −5.45218243618999302573208374452, −4.91461584590121560834030244784, −3.59049683057299957096515029460, −2.91938802305666395474060611604, −1.92487551084706661762631249280, −0.48418247747877377240088489730, 0.48418247747877377240088489730, 1.92487551084706661762631249280, 2.91938802305666395474060611604, 3.59049683057299957096515029460, 4.91461584590121560834030244784, 5.45218243618999302573208374452, 6.67121043329780711788790803197, 6.84567770276326696293066134168, 7.920816652842604096063382956568, 8.911657580914181102711911098388

Graph of the ZZ-function along the critical line