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Properties

Label 2-5760-1.1-c1-0-34
Degree 22
Conductor 57605760
Sign 11
Analytic cond. 45.993845.9938
Root an. cond. 6.781876.78187
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 5.12·7-s + 2·11-s − 5.12·13-s + 1.12·17-s − 5.12·19-s + 5.12·23-s + 25-s − 8.24·29-s − 7.12·31-s + 5.12·35-s + 5.12·37-s + 2·41-s + 6.24·43-s + 13.1·47-s + 19.2·49-s + 10·53-s + 2·55-s + 6·59-s + 2·61-s − 5.12·65-s + 6.24·67-s + 8·71-s − 4.24·73-s + 10.2·77-s − 4.87·79-s − 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.93·7-s + 0.603·11-s − 1.42·13-s + 0.272·17-s − 1.17·19-s + 1.06·23-s + 0.200·25-s − 1.53·29-s − 1.27·31-s + 0.865·35-s + 0.842·37-s + 0.312·41-s + 0.952·43-s + 1.91·47-s + 2.74·49-s + 1.37·53-s + 0.269·55-s + 0.781·59-s + 0.256·61-s − 0.635·65-s + 0.763·67-s + 0.949·71-s − 0.496·73-s + 1.16·77-s − 0.548·79-s − 0.439·83-s + ⋯

Functional equation

Λ(s)=(5760s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5760s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57605760    =    27⋅32⋅52^{7} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 45.993845.9938
Root analytic conductor: 6.781876.78187
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5760, ( :1/2), 1)(2,\ 5760,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7866114362.786611436
L(12)L(\frac12) ≈\approx 2.7866114362.786611436
L(32)L(\frac{3}{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−T 1 - T
good7 1−5.12T+7T2 1 - 5.12T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1+5.12T+13T2 1 + 5.12T + 13T^{2}
17 1−1.12T+17T2 1 - 1.12T + 17T^{2}
19 1+5.12T+19T2 1 + 5.12T + 19T^{2}
23 1−5.12T+23T2 1 - 5.12T + 23T^{2}
29 1+8.24T+29T2 1 + 8.24T + 29T^{2}
31 1+7.12T+31T2 1 + 7.12T + 31T^{2}
37 1−5.12T+37T2 1 - 5.12T + 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1−6.24T+43T2 1 - 6.24T + 43T^{2}
47 1−13.1T+47T2 1 - 13.1T + 47T^{2}
53 1−10T+53T2 1 - 10T + 53T^{2}
59 1−6T+59T2 1 - 6T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−6.24T+67T2 1 - 6.24T + 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+4.24T+73T2 1 + 4.24T + 73T^{2}
79 1+4.87T+79T2 1 + 4.87T + 79T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−10T+97T2 1 - 10T + 97T^{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

−8.001644622231715371082345259487, −7.43286128492331394867783341088, −6.91970303859033905154740306463, −5.67011872211686513557467980031, −5.33631576215886632835076523247, −4.47703493935471993178056520398, −3.92245559356035957251808557618, −2.41364692880760398643922920785, −2.00563147968681177008788447608, −0.918468970184361290672860681967, 0.918468970184361290672860681967, 2.00563147968681177008788447608, 2.41364692880760398643922920785, 3.92245559356035957251808557618, 4.47703493935471993178056520398, 5.33631576215886632835076523247, 5.67011872211686513557467980031, 6.91970303859033905154740306463, 7.43286128492331394867783341088, 8.001644622231715371082345259487

Graph of the ZZ-function along the critical line