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Properties

Label 2-24e2-4.3-c4-0-29
Degree 22
Conductor 576576
Sign ii
Analytic cond. 59.541059.5410
Root an. cond. 7.716287.71628
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.21·5-s − 24.7i·7-s − 108. i·11-s + 55.5·13-s + 463.·17-s + 406. i·19-s + 122. i·23-s − 623.·25-s + 101.·29-s − 1.34e3i·31-s − 30.1i·35-s − 2.65e3·37-s + 1.66e3·41-s − 181. i·43-s − 2.71e3i·47-s + ⋯
L(s)  = 1  + 0.0486·5-s − 0.505i·7-s − 0.899i·11-s + 0.328·13-s + 1.60·17-s + 1.12i·19-s + 0.232i·23-s − 0.997·25-s + 0.120·29-s − 1.39i·31-s − 0.0245i·35-s − 1.94·37-s + 0.991·41-s − 0.0982i·43-s − 1.22i·47-s + ⋯

Functional equation

Λ(s)=(576s/2 ΓC(s) L(s)=(i Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(576s/2 ΓC(s+2) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 576576    =    26⋅322^{6} \cdot 3^{2}
Sign: ii
Analytic conductor: 59.541059.5410
Root analytic conductor: 7.716287.71628
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ576(127,⋅)\chi_{576} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 576, ( :2), i)(2,\ 576,\ (\ :2),\ i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.7714260361.771426036
L(12)L(\frac12) ≈\approx 1.7714260361.771426036
L(3)L(3) 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 1−1.21T+625T2 1 - 1.21T + 625T^{2}
7 1+24.7iT−2.40e3T2 1 + 24.7iT - 2.40e3T^{2}
11 1+108.iT−1.46e4T2 1 + 108. iT - 1.46e4T^{2}
13 1−55.5T+2.85e4T2 1 - 55.5T + 2.85e4T^{2}
17 1−463.T+8.35e4T2 1 - 463.T + 8.35e4T^{2}
19 1−406.iT−1.30e5T2 1 - 406. iT - 1.30e5T^{2}
23 1−122.iT−2.79e5T2 1 - 122. iT - 2.79e5T^{2}
29 1−101.T+7.07e5T2 1 - 101.T + 7.07e5T^{2}
31 1+1.34e3iT−9.23e5T2 1 + 1.34e3iT - 9.23e5T^{2}
37 1+2.65e3T+1.87e6T2 1 + 2.65e3T + 1.87e6T^{2}
41 1−1.66e3T+2.82e6T2 1 - 1.66e3T + 2.82e6T^{2}
43 1+181.iT−3.41e6T2 1 + 181. iT - 3.41e6T^{2}
47 1+2.71e3iT−4.87e6T2 1 + 2.71e3iT - 4.87e6T^{2}
53 1−2.17e3T+7.89e6T2 1 - 2.17e3T + 7.89e6T^{2}
59 1−2.14e3iT−1.21e7T2 1 - 2.14e3iT - 1.21e7T^{2}
61 1+5.23e3T+1.38e7T2 1 + 5.23e3T + 1.38e7T^{2}
67 1−2.81e3iT−2.01e7T2 1 - 2.81e3iT - 2.01e7T^{2}
71 1+8.04e3iT−2.54e7T2 1 + 8.04e3iT - 2.54e7T^{2}
73 1−4.17e3T+2.83e7T2 1 - 4.17e3T + 2.83e7T^{2}
79 1+9.14e3iT−3.89e7T2 1 + 9.14e3iT - 3.89e7T^{2}
83 1+1.02e4iT−4.74e7T2 1 + 1.02e4iT - 4.74e7T^{2}
89 1−2.30e3T+6.27e7T2 1 - 2.30e3T + 6.27e7T^{2}
97 1−2.29e3T+8.85e7T2 1 - 2.29e3T + 8.85e7T^{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

−10.07731082783855730103518448575, −9.025950704329039768557166374803, −8.020532112937324728558985570901, −7.41086507166706237348977553130, −6.05534094963163961055292216253, −5.51004459851188281834858734581, −3.99496041110997517617752882185, −3.29275622202748492389294255875, −1.68937674848484834684362009722, −0.48215990140188257288387630593, 1.19066936311643474382510110944, 2.45676899899810270783051474854, 3.61358053809685049539002720702, 4.87004131526064915889825322818, 5.68275888316339806602421460557, 6.80794237551553730104672105035, 7.64861186894341426552151242364, 8.647808190551609992805273026021, 9.492014479543106375391187004884, 10.27052320160886257953226869227

Graph of the ZZ-function along the critical line