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Properties

Label 2-1472-1.1-c3-0-128
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  + 5.48·3-s + 7.85·5-s + 7.65·7-s + 3.13·9-s + 2.42·11-s − 62.4·13-s + 43.1·15-s − 117.·17-s + 76.2·19-s + 42.0·21-s − 23·23-s − 63.2·25-s − 131.·27-s − 39.2·29-s − 171.·31-s + 13.3·33-s + 60.1·35-s + 280.·37-s − 342.·39-s − 280.·41-s − 393.·43-s + 24.6·45-s + 467.·47-s − 284.·49-s − 642.·51-s − 253.·53-s + 19.0·55-s + ⋯
L(s)  = 1  + 1.05·3-s + 0.702·5-s + 0.413·7-s + 0.116·9-s + 0.0664·11-s − 1.33·13-s + 0.742·15-s − 1.66·17-s + 0.920·19-s + 0.436·21-s − 0.208·23-s − 0.506·25-s − 0.933·27-s − 0.251·29-s − 0.995·31-s + 0.0702·33-s + 0.290·35-s + 1.24·37-s − 1.40·39-s − 1.06·41-s − 1.39·43-s + 0.0816·45-s + 1.45·47-s − 0.829·49-s − 1.76·51-s − 0.656·53-s + 0.0467·55-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 1−5.48T+27T2 1 - 5.48T + 27T^{2}
5 1−7.85T+125T2 1 - 7.85T + 125T^{2}
7 1−7.65T+343T2 1 - 7.65T + 343T^{2}
11 1−2.42T+1.33e3T2 1 - 2.42T + 1.33e3T^{2}
13 1+62.4T+2.19e3T2 1 + 62.4T + 2.19e3T^{2}
17 1+117.T+4.91e3T2 1 + 117.T + 4.91e3T^{2}
19 1−76.2T+6.85e3T2 1 - 76.2T + 6.85e3T^{2}
29 1+39.2T+2.43e4T2 1 + 39.2T + 2.43e4T^{2}
31 1+171.T+2.97e4T2 1 + 171.T + 2.97e4T^{2}
37 1−280.T+5.06e4T2 1 - 280.T + 5.06e4T^{2}
41 1+280.T+6.89e4T2 1 + 280.T + 6.89e4T^{2}
43 1+393.T+7.95e4T2 1 + 393.T + 7.95e4T^{2}
47 1−467.T+1.03e5T2 1 - 467.T + 1.03e5T^{2}
53 1+253.T+1.48e5T2 1 + 253.T + 1.48e5T^{2}
59 1+850.T+2.05e5T2 1 + 850.T + 2.05e5T^{2}
61 1+176.T+2.26e5T2 1 + 176.T + 2.26e5T^{2}
67 1−684.T+3.00e5T2 1 - 684.T + 3.00e5T^{2}
71 1−1.11e3T+3.57e5T2 1 - 1.11e3T + 3.57e5T^{2}
73 1+510.T+3.89e5T2 1 + 510.T + 3.89e5T^{2}
79 1+535.T+4.93e5T2 1 + 535.T + 4.93e5T^{2}
83 1+323.T+5.71e5T2 1 + 323.T + 5.71e5T^{2}
89 1−327.T+7.04e5T2 1 - 327.T + 7.04e5T^{2}
97 1−1.65e3T+9.12e5T2 1 - 1.65e3T + 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

−8.864969754372233457840796204721, −7.949974610440991867566653211126, −7.32812802138215114433187846550, −6.33583881540520041636291417471, −5.31951813621714904934739771577, −4.50889868843394077189249043663, −3.35774937188101063063408846775, −2.36628810954434616002290239018, −1.80421188699432600531460994462, 0, 1.80421188699432600531460994462, 2.36628810954434616002290239018, 3.35774937188101063063408846775, 4.50889868843394077189249043663, 5.31951813621714904934739771577, 6.33583881540520041636291417471, 7.32812802138215114433187846550, 7.949974610440991867566653211126, 8.864969754372233457840796204721

Graph of the ZZ-function along the critical line