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Properties

Label 2-1088-1.1-c3-0-69
Degree 22
Conductor 10881088
Sign −1-1
Analytic cond. 64.194064.1940
Root an. cond. 8.012128.01212
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  + 1.46·3-s − 0.928·5-s − 7.60·7-s − 24.8·9-s − 35.0·11-s + 83.5·13-s − 1.35·15-s + 17·17-s + 115.·19-s − 11.1·21-s − 64.6·23-s − 124.·25-s − 75.9·27-s + 288.·29-s + 186.·31-s − 51.2·33-s + 7.06·35-s − 241.·37-s + 122.·39-s − 43.8·41-s − 257.·43-s + 23.0·45-s − 40.1·47-s − 285.·49-s + 24.8·51-s − 214.·53-s + 32.5·55-s + ⋯
L(s)  = 1  + 0.281·3-s − 0.0830·5-s − 0.410·7-s − 0.920·9-s − 0.960·11-s + 1.78·13-s − 0.0233·15-s + 0.242·17-s + 1.39·19-s − 0.115·21-s − 0.585·23-s − 0.993·25-s − 0.541·27-s + 1.84·29-s + 1.07·31-s − 0.270·33-s + 0.0341·35-s − 1.07·37-s + 0.502·39-s − 0.167·41-s − 0.913·43-s + 0.0764·45-s − 0.124·47-s − 0.831·49-s + 0.0683·51-s − 0.557·53-s + 0.0797·55-s + ⋯

Functional equation

Λ(s)=(1088s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1088s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10881088    =    26⋅172^{6} \cdot 17
Sign: −1-1
Analytic conductor: 64.194064.1940
Root analytic conductor: 8.012128.01212
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1088, ( :3/2), −1)(2,\ 1088,\ (\ :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
17 1−17T 1 - 17T
good3 1−1.46T+27T2 1 - 1.46T + 27T^{2}
5 1+0.928T+125T2 1 + 0.928T + 125T^{2}
7 1+7.60T+343T2 1 + 7.60T + 343T^{2}
11 1+35.0T+1.33e3T2 1 + 35.0T + 1.33e3T^{2}
13 1−83.5T+2.19e3T2 1 - 83.5T + 2.19e3T^{2}
19 1−115.T+6.85e3T2 1 - 115.T + 6.85e3T^{2}
23 1+64.6T+1.21e4T2 1 + 64.6T + 1.21e4T^{2}
29 1−288.T+2.43e4T2 1 - 288.T + 2.43e4T^{2}
31 1−186.T+2.97e4T2 1 - 186.T + 2.97e4T^{2}
37 1+241.T+5.06e4T2 1 + 241.T + 5.06e4T^{2}
41 1+43.8T+6.89e4T2 1 + 43.8T + 6.89e4T^{2}
43 1+257.T+7.95e4T2 1 + 257.T + 7.95e4T^{2}
47 1+40.1T+1.03e5T2 1 + 40.1T + 1.03e5T^{2}
53 1+214.T+1.48e5T2 1 + 214.T + 1.48e5T^{2}
59 1+585.T+2.05e5T2 1 + 585.T + 2.05e5T^{2}
61 1+331.T+2.26e5T2 1 + 331.T + 2.26e5T^{2}
67 1+819.T+3.00e5T2 1 + 819.T + 3.00e5T^{2}
71 1+961.T+3.57e5T2 1 + 961.T + 3.57e5T^{2}
73 1+342.T+3.89e5T2 1 + 342.T + 3.89e5T^{2}
79 1+513.T+4.93e5T2 1 + 513.T + 4.93e5T^{2}
83 1+270.T+5.71e5T2 1 + 270.T + 5.71e5T^{2}
89 1−728.T+7.04e5T2 1 - 728.T + 7.04e5T^{2}
97 1−211.T+9.12e5T2 1 - 211.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

−8.922723032500498994338663390832, −8.276143122540459636571634658689, −7.65085262969733297581876114077, −6.34011601052226332631537881477, −5.81473190183912245583735541221, −4.74813342734349494891691899977, −3.40449090200445055138832119392, −2.93152195914184346028677183555, −1.40304645234449631957290531624, 0, 1.40304645234449631957290531624, 2.93152195914184346028677183555, 3.40449090200445055138832119392, 4.74813342734349494891691899977, 5.81473190183912245583735541221, 6.34011601052226332631537881477, 7.65085262969733297581876114077, 8.276143122540459636571634658689, 8.922723032500498994338663390832

Graph of the ZZ-function along the critical line