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Properties

Label 2-1472-1.1-c3-0-89
Degree 22
Conductor 14721472
Sign −1-1
Analytic cond. 86.850886.8508
Root an. cond. 9.319379.31937
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.55·3-s + 15.7·5-s − 25.1·7-s − 20.4·9-s − 15.8·11-s − 15.4·13-s − 40.2·15-s + 56.7·17-s + 107.·19-s + 64.4·21-s − 23·23-s + 122.·25-s + 121.·27-s + 267.·29-s + 35.0·31-s + 40.5·33-s − 396.·35-s − 84.9·37-s + 39.6·39-s + 296.·41-s − 353.·43-s − 322.·45-s − 86.6·47-s + 291.·49-s − 145.·51-s + 126.·53-s − 249.·55-s + ⋯
L(s)  = 1  − 0.492·3-s + 1.40·5-s − 1.35·7-s − 0.757·9-s − 0.434·11-s − 0.330·13-s − 0.693·15-s + 0.810·17-s + 1.29·19-s + 0.669·21-s − 0.208·23-s + 0.982·25-s + 0.865·27-s + 1.71·29-s + 0.203·31-s + 0.213·33-s − 1.91·35-s − 0.377·37-s + 0.162·39-s + 1.13·41-s − 1.25·43-s − 1.06·45-s − 0.268·47-s + 0.849·49-s − 0.398·51-s + 0.328·53-s − 0.611·55-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: −1-1
Analytic conductor: 86.850886.8508
Root analytic conductor: 9.319379.31937
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1472, ( :3/2), −1)(2,\ 1472,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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
23 1+23T 1 + 23T
good3 1+2.55T+27T2 1 + 2.55T + 27T^{2}
5 1−15.7T+125T2 1 - 15.7T + 125T^{2}
7 1+25.1T+343T2 1 + 25.1T + 343T^{2}
11 1+15.8T+1.33e3T2 1 + 15.8T + 1.33e3T^{2}
13 1+15.4T+2.19e3T2 1 + 15.4T + 2.19e3T^{2}
17 1−56.7T+4.91e3T2 1 - 56.7T + 4.91e3T^{2}
19 1−107.T+6.85e3T2 1 - 107.T + 6.85e3T^{2}
29 1−267.T+2.43e4T2 1 - 267.T + 2.43e4T^{2}
31 1−35.0T+2.97e4T2 1 - 35.0T + 2.97e4T^{2}
37 1+84.9T+5.06e4T2 1 + 84.9T + 5.06e4T^{2}
41 1−296.T+6.89e4T2 1 - 296.T + 6.89e4T^{2}
43 1+353.T+7.95e4T2 1 + 353.T + 7.95e4T^{2}
47 1+86.6T+1.03e5T2 1 + 86.6T + 1.03e5T^{2}
53 1−126.T+1.48e5T2 1 - 126.T + 1.48e5T^{2}
59 1+853.T+2.05e5T2 1 + 853.T + 2.05e5T^{2}
61 1+647.T+2.26e5T2 1 + 647.T + 2.26e5T^{2}
67 1+603.T+3.00e5T2 1 + 603.T + 3.00e5T^{2}
71 1+467.T+3.57e5T2 1 + 467.T + 3.57e5T^{2}
73 1−301.T+3.89e5T2 1 - 301.T + 3.89e5T^{2}
79 1−766.T+4.93e5T2 1 - 766.T + 4.93e5T^{2}
83 1+660.T+5.71e5T2 1 + 660.T + 5.71e5T^{2}
89 1+1.10e3T+7.04e5T2 1 + 1.10e3T + 7.04e5T^{2}
97 1+1.49e3T+9.12e5T2 1 + 1.49e3T + 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

−9.000991389657237037138956752145, −7.915817742651554666104812337060, −6.83544723988830596699057426821, −6.10930266451819008948164600170, −5.64394925923888431050723551809, −4.82555452758901574163683804352, −3.15555996760764495784993465668, −2.72668353865807251308029044285, −1.25033350857530458615822580454, 0, 1.25033350857530458615822580454, 2.72668353865807251308029044285, 3.15555996760764495784993465668, 4.82555452758901574163683804352, 5.64394925923888431050723551809, 6.10930266451819008948164600170, 6.83544723988830596699057426821, 7.915817742651554666104812337060, 9.000991389657237037138956752145

Graph of the ZZ-function along the critical line