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Properties

Label 2-288-8.5-c3-0-12
Degree 22
Conductor 288288
Sign ii
Analytic cond. 16.992516.9925
Root an. cond. 4.122204.12220
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 19.7i·5-s + 34·7-s − 5.65i·11-s − 267·25-s − 223. i·29-s + 70·31-s − 673. i·35-s + 813·49-s − 579. i·53-s − 112.·55-s + 554. i·59-s − 322·73-s − 192. i·77-s − 1.37e3·79-s − 1.22e3i·83-s + ⋯
L(s)  = 1  − 1.77i·5-s + 1.83·7-s − 0.155i·11-s − 2.13·25-s − 1.43i·29-s + 0.405·31-s − 3.25i·35-s + 2.37·49-s − 1.50i·53-s − 0.274·55-s + 1.22i·59-s − 0.516·73-s − 0.284i·77-s − 1.95·79-s − 1.62i·83-s + ⋯

Functional equation

Λ(s)=(288s/2 ΓC(s) L(s)=(i Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(288s/2 ΓC(s+3/2) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 288288    =    25⋅322^{5} \cdot 3^{2}
Sign: ii
Analytic conductor: 16.992516.9925
Root analytic conductor: 4.122204.12220
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ288(145,⋅)\chi_{288} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 288, ( :3/2), i)(2,\ 288,\ (\ :3/2),\ i)

Particular Values

L(2)L(2) ≈\approx 2.1089196872.108919687
L(12)L(\frac12) ≈\approx 2.1089196872.108919687
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
3 1 1
good5 1+19.7iT−125T2 1 + 19.7iT - 125T^{2}
7 1−34T+343T2 1 - 34T + 343T^{2}
11 1+5.65iT−1.33e3T2 1 + 5.65iT - 1.33e3T^{2}
13 1−2.19e3T2 1 - 2.19e3T^{2}
17 1+4.91e3T2 1 + 4.91e3T^{2}
19 1−6.85e3T2 1 - 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1+223.iT−2.43e4T2 1 + 223. iT - 2.43e4T^{2}
31 1−70T+2.97e4T2 1 - 70T + 2.97e4T^{2}
37 1−5.06e4T2 1 - 5.06e4T^{2}
41 1+6.89e4T2 1 + 6.89e4T^{2}
43 1−7.95e4T2 1 - 7.95e4T^{2}
47 1+1.03e5T2 1 + 1.03e5T^{2}
53 1+579.iT−1.48e5T2 1 + 579. iT - 1.48e5T^{2}
59 1−554.iT−2.05e5T2 1 - 554. iT - 2.05e5T^{2}
61 1−2.26e5T2 1 - 2.26e5T^{2}
67 1−3.00e5T2 1 - 3.00e5T^{2}
71 1+3.57e5T2 1 + 3.57e5T^{2}
73 1+322T+3.89e5T2 1 + 322T + 3.89e5T^{2}
79 1+1.37e3T+4.93e5T2 1 + 1.37e3T + 4.93e5T^{2}
83 1+1.22e3iT−5.71e5T2 1 + 1.22e3iT - 5.71e5T^{2}
89 1+7.04e5T2 1 + 7.04e5T^{2}
97 1+574T+9.12e5T2 1 + 574T + 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

−11.48214944191593294086476533915, −10.15296807908926188882202869298, −8.955815583917076791416776789525, −8.334164412457794528064257586476, −7.63005488804058618439352639941, −5.79052724918491123696966831234, −4.89951563441063662155692358808, −4.23736571867318702157920557289, −1.93407725571111384367408329568, −0.854692854043478836273477347845, 1.73220427893793231135780452322, 2.97176518867559875544422242109, 4.38150302547969145180981064832, 5.62391519078560818375936434318, 6.89675995251773808252489858093, 7.60638010169990584021963497024, 8.573577037459673877182630969198, 10.00044265947340884484848528676, 10.93236559115647659812227445442, 11.24228682408501507032598225570

Graph of the ZZ-function along the critical line