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Properties

Label 2-2160-1.1-c3-0-51
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 + 34.6·7-s + 23.2·11-s + 60.0·13-s − 19.6·17-s + 10.2·19-s + 79.7·23-s + 25·25-s − 110.·29-s + 42.5·31-s + 173.·35-s + 308.·37-s + 106.·41-s + 467.·43-s + 37.7·47-s + 858.·49-s + 568.·53-s + 116.·55-s − 666.·59-s − 862.·61-s + 300.·65-s − 547.·67-s − 761.·71-s − 216.·73-s + 805.·77-s − 258.·79-s − 903.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.87·7-s + 0.636·11-s + 1.28·13-s − 0.280·17-s + 0.123·19-s + 0.723·23-s + 0.200·25-s − 0.709·29-s + 0.246·31-s + 0.837·35-s + 1.37·37-s + 0.407·41-s + 1.65·43-s + 0.117·47-s + 2.50·49-s + 1.47·53-s + 0.284·55-s − 1.47·59-s − 1.80·61-s + 0.573·65-s − 0.997·67-s − 1.27·71-s − 0.347·73-s + 1.19·77-s − 0.368·79-s − 1.19·83-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 4.0202739084.020273908
L(12)L(\frac12) ≈\approx 4.0202739084.020273908
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 1−34.6T+343T2 1 - 34.6T + 343T^{2}
11 1−23.2T+1.33e3T2 1 - 23.2T + 1.33e3T^{2}
13 1−60.0T+2.19e3T2 1 - 60.0T + 2.19e3T^{2}
17 1+19.6T+4.91e3T2 1 + 19.6T + 4.91e3T^{2}
19 1−10.2T+6.85e3T2 1 - 10.2T + 6.85e3T^{2}
23 1−79.7T+1.21e4T2 1 - 79.7T + 1.21e4T^{2}
29 1+110.T+2.43e4T2 1 + 110.T + 2.43e4T^{2}
31 1−42.5T+2.97e4T2 1 - 42.5T + 2.97e4T^{2}
37 1−308.T+5.06e4T2 1 - 308.T + 5.06e4T^{2}
41 1−106.T+6.89e4T2 1 - 106.T + 6.89e4T^{2}
43 1−467.T+7.95e4T2 1 - 467.T + 7.95e4T^{2}
47 1−37.7T+1.03e5T2 1 - 37.7T + 1.03e5T^{2}
53 1−568.T+1.48e5T2 1 - 568.T + 1.48e5T^{2}
59 1+666.T+2.05e5T2 1 + 666.T + 2.05e5T^{2}
61 1+862.T+2.26e5T2 1 + 862.T + 2.26e5T^{2}
67 1+547.T+3.00e5T2 1 + 547.T + 3.00e5T^{2}
71 1+761.T+3.57e5T2 1 + 761.T + 3.57e5T^{2}
73 1+216.T+3.89e5T2 1 + 216.T + 3.89e5T^{2}
79 1+258.T+4.93e5T2 1 + 258.T + 4.93e5T^{2}
83 1+903.T+5.71e5T2 1 + 903.T + 5.71e5T^{2}
89 1−1.26e3T+7.04e5T2 1 - 1.26e3T + 7.04e5T^{2}
97 1−617.T+9.12e5T2 1 - 617.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

−8.955621719697950988937508159257, −7.905599923538882828231063251433, −7.35988822784565074292177760372, −6.18797086675618530305834167098, −5.63471595729629378866781393814, −4.60738949418755871802913163234, −4.05420387706969139084496259359, −2.71282777438757283590115649406, −1.60026692718911216432123797919, −1.04705231529672536272627922277, 1.04705231529672536272627922277, 1.60026692718911216432123797919, 2.71282777438757283590115649406, 4.05420387706969139084496259359, 4.60738949418755871802913163234, 5.63471595729629378866781393814, 6.18797086675618530305834167098, 7.35988822784565074292177760372, 7.905599923538882828231063251433, 8.955621719697950988937508159257

Graph of the ZZ-function along the critical line