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Properties

Label 2-1472-1.1-c3-0-33
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  − 7.87·3-s − 13.0·5-s − 35.1·7-s + 35.0·9-s − 66.4·11-s + 40.5·13-s + 103.·15-s + 39.8·17-s − 93.1·19-s + 276.·21-s − 23·23-s + 46.1·25-s − 63.0·27-s − 234.·29-s + 226.·31-s + 523.·33-s + 459.·35-s + 275.·37-s − 318.·39-s + 59.0·41-s − 184.·43-s − 457.·45-s + 506.·47-s + 889.·49-s − 313.·51-s + 8.25·53-s + 869.·55-s + ⋯
L(s)  = 1  − 1.51·3-s − 1.17·5-s − 1.89·7-s + 1.29·9-s − 1.82·11-s + 0.864·13-s + 1.77·15-s + 0.568·17-s − 1.12·19-s + 2.87·21-s − 0.208·23-s + 0.369·25-s − 0.449·27-s − 1.50·29-s + 1.31·31-s + 2.76·33-s + 2.21·35-s + 1.22·37-s − 1.30·39-s + 0.224·41-s − 0.652·43-s − 1.51·45-s + 1.57·47-s + 2.59·49-s − 0.861·51-s + 0.0213·53-s + 2.13·55-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+7.87T+27T2 1 + 7.87T + 27T^{2}
5 1+13.0T+125T2 1 + 13.0T + 125T^{2}
7 1+35.1T+343T2 1 + 35.1T + 343T^{2}
11 1+66.4T+1.33e3T2 1 + 66.4T + 1.33e3T^{2}
13 1−40.5T+2.19e3T2 1 - 40.5T + 2.19e3T^{2}
17 1−39.8T+4.91e3T2 1 - 39.8T + 4.91e3T^{2}
19 1+93.1T+6.85e3T2 1 + 93.1T + 6.85e3T^{2}
29 1+234.T+2.43e4T2 1 + 234.T + 2.43e4T^{2}
31 1−226.T+2.97e4T2 1 - 226.T + 2.97e4T^{2}
37 1−275.T+5.06e4T2 1 - 275.T + 5.06e4T^{2}
41 1−59.0T+6.89e4T2 1 - 59.0T + 6.89e4T^{2}
43 1+184.T+7.95e4T2 1 + 184.T + 7.95e4T^{2}
47 1−506.T+1.03e5T2 1 - 506.T + 1.03e5T^{2}
53 1−8.25T+1.48e5T2 1 - 8.25T + 1.48e5T^{2}
59 1−483.T+2.05e5T2 1 - 483.T + 2.05e5T^{2}
61 1+411.T+2.26e5T2 1 + 411.T + 2.26e5T^{2}
67 1+324.T+3.00e5T2 1 + 324.T + 3.00e5T^{2}
71 1−674.T+3.57e5T2 1 - 674.T + 3.57e5T^{2}
73 1+752.T+3.89e5T2 1 + 752.T + 3.89e5T^{2}
79 1+1.12e3T+4.93e5T2 1 + 1.12e3T + 4.93e5T^{2}
83 1−136.T+5.71e5T2 1 - 136.T + 5.71e5T^{2}
89 1+944.T+7.04e5T2 1 + 944.T + 7.04e5T^{2}
97 1−271.T+9.12e5T2 1 - 271.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

−8.669910736923300202928062480155, −7.73399009004740602842935110567, −7.02265604759270307343278668882, −6.03726470406550353149570859634, −5.75083290751400005618936916304, −4.51309614775891442840290743931, −3.68627005356804579590516419731, −2.69766523348999238617496922337, −0.62261733248286535705815709966, 0, 0.62261733248286535705815709966, 2.69766523348999238617496922337, 3.68627005356804579590516419731, 4.51309614775891442840290743931, 5.75083290751400005618936916304, 6.03726470406550353149570859634, 7.02265604759270307343278668882, 7.73399009004740602842935110567, 8.669910736923300202928062480155

Graph of the ZZ-function along the critical line