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Properties

Label 2-24e2-8.3-c4-0-2
Degree 22
Conductor 576576
Sign −0.707−0.707i-0.707 - 0.707i
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  − 24.2i·5-s + 58i·7-s + 13.8·11-s − 20.7i·13-s + 306·17-s − 602.·19-s + 468i·23-s + 37·25-s − 1.46e3i·29-s + 110i·31-s + 1.40e3·35-s + 1.03e3i·37-s − 2.97e3·41-s − 2.88e3·43-s − 396i·47-s + ⋯
L(s)  = 1  − 0.969i·5-s + 1.18i·7-s + 0.114·11-s − 0.122i·13-s + 1.05·17-s − 1.66·19-s + 0.884i·23-s + 0.0592·25-s − 1.74i·29-s + 0.114i·31-s + 1.14·35-s + 0.759i·37-s − 1.76·41-s − 1.56·43-s − 0.179i·47-s + ⋯

Functional equation

Λ(s)=(576s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(576s/2 ΓC(s+2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 576576    =    26⋅322^{6} \cdot 3^{2}
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 59.541059.5410
Root analytic conductor: 7.716287.71628
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ576(415,⋅)\chi_{576} (415, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 576, ( :2), −0.707−0.707i)(2,\ 576,\ (\ :2),\ -0.707 - 0.707i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.64592266420.6459226642
L(12)L(\frac12) ≈\approx 0.64592266420.6459226642
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+24.2iT−625T2 1 + 24.2iT - 625T^{2}
7 1−58iT−2.40e3T2 1 - 58iT - 2.40e3T^{2}
11 1−13.8T+1.46e4T2 1 - 13.8T + 1.46e4T^{2}
13 1+20.7iT−2.85e4T2 1 + 20.7iT - 2.85e4T^{2}
17 1−306T+8.35e4T2 1 - 306T + 8.35e4T^{2}
19 1+602.T+1.30e5T2 1 + 602.T + 1.30e5T^{2}
23 1−468iT−2.79e5T2 1 - 468iT - 2.79e5T^{2}
29 1+1.46e3iT−7.07e5T2 1 + 1.46e3iT - 7.07e5T^{2}
31 1−110iT−9.23e5T2 1 - 110iT - 9.23e5T^{2}
37 1−1.03e3iT−1.87e6T2 1 - 1.03e3iT - 1.87e6T^{2}
41 1+2.97e3T+2.82e6T2 1 + 2.97e3T + 2.82e6T^{2}
43 1+2.88e3T+3.41e6T2 1 + 2.88e3T + 3.41e6T^{2}
47 1+396iT−4.87e6T2 1 + 396iT - 4.87e6T^{2}
53 1+1.12e3iT−7.89e6T2 1 + 1.12e3iT - 7.89e6T^{2}
59 1−2.68e3T+1.21e7T2 1 - 2.68e3T + 1.21e7T^{2}
61 1−5.98e3iT−1.38e7T2 1 - 5.98e3iT - 1.38e7T^{2}
67 1−4.80e3T+2.01e7T2 1 - 4.80e3T + 2.01e7T^{2}
71 1−6.58e3iT−2.54e7T2 1 - 6.58e3iT - 2.54e7T^{2}
73 1+5.89e3T+2.83e7T2 1 + 5.89e3T + 2.83e7T^{2}
79 1−8.48e3iT−3.89e7T2 1 - 8.48e3iT - 3.89e7T^{2}
83 1+13.8T+4.74e7T2 1 + 13.8T + 4.74e7T^{2}
89 1+8.76e3T+6.27e7T2 1 + 8.76e3T + 6.27e7T^{2}
97 1−5.91e3T+8.85e7T2 1 - 5.91e3T + 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

−10.25956414003880706820223437073, −9.540739957788682049875974146481, −8.481937147047162956721966538713, −8.253645381392243787251220605178, −6.74935382347212281887900576114, −5.71707568254131179888104616009, −5.05039813869849369966718501602, −3.87807082957901220294997195080, −2.50603213485441464834279846211, −1.34453268586427533506611922395, 0.15988433332669184288079933666, 1.64889686292710059698867690061, 3.07772995700007501562644237282, 3.93296366845340768884598355175, 5.06387795348541500967621120396, 6.52685681212796169166746256605, 6.89926591036386455210925821746, 7.926568577177837480107636921226, 8.869195125078180814517429853246, 10.26132527961587393357509992981

Graph of the ZZ-function along the critical line