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Properties

Label 2-24e2-8.5-c7-0-52
Degree 22
Conductor 576576
Sign −0.258+0.965i-0.258 + 0.965i
Analytic cond. 179.933179.933
Root an. cond. 13.413913.4139
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 435. i·5-s − 34.1·7-s + 3.88e3i·11-s − 2.56e3i·13-s + 2.32e4·17-s − 2.38e4i·19-s − 6.04e4·23-s − 1.11e5·25-s − 3.93e4i·29-s − 7.04e4·31-s − 1.48e4i·35-s + 2.93e5i·37-s − 3.77e5·41-s + 1.01e6i·43-s + 2.19e5·47-s + ⋯
L(s)  = 1  + 1.55i·5-s − 0.0376·7-s + 0.879i·11-s − 0.323i·13-s + 1.14·17-s − 0.797i·19-s − 1.03·23-s − 1.43·25-s − 0.299i·29-s − 0.424·31-s − 0.0586i·35-s + 0.954i·37-s − 0.855·41-s + 1.95i·43-s + 0.308·47-s + ⋯

Functional equation

Λ(s)=(576s/2 ΓC(s) L(s)=((−0.258+0.965i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(576s/2 ΓC(s+7/2) L(s)=((−0.258+0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 576576    =    26⋅322^{6} \cdot 3^{2}
Sign: −0.258+0.965i-0.258 + 0.965i
Analytic conductor: 179.933179.933
Root analytic conductor: 13.413913.4139
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ576(289,⋅)\chi_{576} (289, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 576, ( :7/2), −0.258+0.965i)(2,\ 576,\ (\ :7/2),\ -0.258 + 0.965i)

Particular Values

L(4)L(4) ≈\approx 0.054744401440.05474440144
L(12)L(\frac12) ≈\approx 0.054744401440.05474440144
L(92)L(\frac{9}{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 1−435.iT−7.81e4T2 1 - 435. iT - 7.81e4T^{2}
7 1+34.1T+8.23e5T2 1 + 34.1T + 8.23e5T^{2}
11 1−3.88e3iT−1.94e7T2 1 - 3.88e3iT - 1.94e7T^{2}
13 1+2.56e3iT−6.27e7T2 1 + 2.56e3iT - 6.27e7T^{2}
17 1−2.32e4T+4.10e8T2 1 - 2.32e4T + 4.10e8T^{2}
19 1+2.38e4iT−8.93e8T2 1 + 2.38e4iT - 8.93e8T^{2}
23 1+6.04e4T+3.40e9T2 1 + 6.04e4T + 3.40e9T^{2}
29 1+3.93e4iT−1.72e10T2 1 + 3.93e4iT - 1.72e10T^{2}
31 1+7.04e4T+2.75e10T2 1 + 7.04e4T + 2.75e10T^{2}
37 1−2.93e5iT−9.49e10T2 1 - 2.93e5iT - 9.49e10T^{2}
41 1+3.77e5T+1.94e11T2 1 + 3.77e5T + 1.94e11T^{2}
43 1−1.01e6iT−2.71e11T2 1 - 1.01e6iT - 2.71e11T^{2}
47 1−2.19e5T+5.06e11T2 1 - 2.19e5T + 5.06e11T^{2}
53 1+1.74e6iT−1.17e12T2 1 + 1.74e6iT - 1.17e12T^{2}
59 1−2.40e6iT−2.48e12T2 1 - 2.40e6iT - 2.48e12T^{2}
61 1+2.75e6iT−3.14e12T2 1 + 2.75e6iT - 3.14e12T^{2}
67 1+1.00e6iT−6.06e12T2 1 + 1.00e6iT - 6.06e12T^{2}
71 1−1.53e6T+9.09e12T2 1 - 1.53e6T + 9.09e12T^{2}
73 1+6.06e6T+1.10e13T2 1 + 6.06e6T + 1.10e13T^{2}
79 1+5.69e6T+1.92e13T2 1 + 5.69e6T + 1.92e13T^{2}
83 1−4.17e6iT−2.71e13T2 1 - 4.17e6iT - 2.71e13T^{2}
89 1+3.89e5T+4.42e13T2 1 + 3.89e5T + 4.42e13T^{2}
97 1−2.87e6T+8.07e13T2 1 - 2.87e6T + 8.07e13T^{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

−9.827689026708452889554945140567, −8.253188044494481725970057358501, −7.44938022497856911238042974345, −6.72906186240479914107262903661, −5.88313251121319706998782013263, −4.67137669235170673607711975924, −3.45037274438999891398822731570, −2.72705855728268404904116596905, −1.62242120913369670031461205016, −0.01077099369620408158576725613, 0.981416756164718725134344947955, 1.84441984677747658607421219456, 3.43320209143751308813407117148, 4.28965145226464887971017486002, 5.43759555286042420926168409528, 5.89458537532686087055822316341, 7.37797181993526185716293726175, 8.292033897736729514401547287746, 8.845142615071568624462643439989, 9.736700828949141378818274104489

Graph of the ZZ-function along the critical line