↑

Properties

Label 2-24e2-8.5-c7-0-27
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 214. i·5-s + 616.·7-s + 5.12e3i·11-s − 39.5i·13-s − 3.77e3·17-s − 2.12e4i·19-s + 2.41e4·23-s + 3.22e4·25-s + 1.64e5i·29-s + 2.14e5·31-s + 1.32e5i·35-s − 7.67e4i·37-s + 2.08e5·41-s + 5.96e5i·43-s + 3.03e5·47-s + ⋯
L(s)  = 1  + 0.766i·5-s + 0.678·7-s + 1.16i·11-s − 0.00499i·13-s − 0.186·17-s − 0.709i·19-s + 0.413·23-s + 0.412·25-s + 1.25i·29-s + 1.29·31-s + 0.520i·35-s − 0.249i·37-s + 0.471·41-s + 1.14i·43-s + 0.427·47-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 2.3607037942.360703794
L(12)L(\frac12) ≈\approx 2.3607037942.360703794
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 1−214.iT−7.81e4T2 1 - 214. iT - 7.81e4T^{2}
7 1−616.T+8.23e5T2 1 - 616.T + 8.23e5T^{2}
11 1−5.12e3iT−1.94e7T2 1 - 5.12e3iT - 1.94e7T^{2}
13 1+39.5iT−6.27e7T2 1 + 39.5iT - 6.27e7T^{2}
17 1+3.77e3T+4.10e8T2 1 + 3.77e3T + 4.10e8T^{2}
19 1+2.12e4iT−8.93e8T2 1 + 2.12e4iT - 8.93e8T^{2}
23 1−2.41e4T+3.40e9T2 1 - 2.41e4T + 3.40e9T^{2}
29 1−1.64e5iT−1.72e10T2 1 - 1.64e5iT - 1.72e10T^{2}
31 1−2.14e5T+2.75e10T2 1 - 2.14e5T + 2.75e10T^{2}
37 1+7.67e4iT−9.49e10T2 1 + 7.67e4iT - 9.49e10T^{2}
41 1−2.08e5T+1.94e11T2 1 - 2.08e5T + 1.94e11T^{2}
43 1−5.96e5iT−2.71e11T2 1 - 5.96e5iT - 2.71e11T^{2}
47 1−3.03e5T+5.06e11T2 1 - 3.03e5T + 5.06e11T^{2}
53 1−7.73e4iT−1.17e12T2 1 - 7.73e4iT - 1.17e12T^{2}
59 1−6.22e5iT−2.48e12T2 1 - 6.22e5iT - 2.48e12T^{2}
61 1+2.89e6iT−3.14e12T2 1 + 2.89e6iT - 3.14e12T^{2}
67 1−3.20e6iT−6.06e12T2 1 - 3.20e6iT - 6.06e12T^{2}
71 1−4.09e6T+9.09e12T2 1 - 4.09e6T + 9.09e12T^{2}
73 1−3.03e6T+1.10e13T2 1 - 3.03e6T + 1.10e13T^{2}
79 1+4.74e6T+1.92e13T2 1 + 4.74e6T + 1.92e13T^{2}
83 1+6.52e6iT−2.71e13T2 1 + 6.52e6iT - 2.71e13T^{2}
89 1−1.08e7T+4.42e13T2 1 - 1.08e7T + 4.42e13T^{2}
97 1+1.10e7T+8.07e13T2 1 + 1.10e7T + 8.07e13T^{2}
show more
show less
   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

−9.910537753007487126679319324734, −9.017653295851025891194346335110, −7.989891518082642884851406550525, −7.11403316743546650710560777272, −6.49291035830447366843627636156, −5.10837709430538833911370839989, −4.43774350614545811787220734872, −3.09679608964965850092848549217, −2.19097246942979421054913178777, −1.05810400553615557685768833790, 0.49578418999686421563825969509, 1.27891714276018275772213092128, 2.53917011496576101443087458143, 3.79113884292515341053720765770, 4.76346555213741410128226635683, 5.60969443803752932696118706462, 6.53488151714982351103195491705, 7.86786045658766416303554168276, 8.402104867786523011649666409234, 9.180512908911820280444085135011

Graph of the ZZ-function along the critical line