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Properties

Label 2-2e3-8.5-c21-0-17
Degree 22
Conductor 88
Sign −0.711+0.702i-0.711 + 0.702i
Analytic cond. 22.358122.3581
Root an. cond. 4.728444.72844
Motivic weight 2121
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−377. + 1.39e3i)2-s − 1.48e5i·3-s + (−1.81e6 − 1.05e6i)4-s − 2.51e7i·5-s + (2.07e8 + 5.60e7i)6-s + 8.96e8·7-s + (2.16e9 − 2.13e9i)8-s − 1.15e10·9-s + (3.51e10 + 9.51e9i)10-s − 1.46e11i·11-s + (−1.56e11 + 2.68e11i)12-s + 4.53e11i·13-s + (−3.38e11 + 1.25e12i)14-s − 3.73e12·15-s + (2.16e12 + 3.82e12i)16-s + 6.08e11·17-s + ⋯
L(s)  = 1  + (−0.260 + 0.965i)2-s − 1.44i·3-s + (−0.863 − 0.503i)4-s − 1.15i·5-s + (1.39 + 0.378i)6-s + 1.19·7-s + (0.711 − 0.702i)8-s − 1.10·9-s + (1.11 + 0.300i)10-s − 1.70i·11-s + (−0.730 + 1.25i)12-s + 0.913i·13-s + (−0.312 + 1.15i)14-s − 1.67·15-s + (0.492 + 0.870i)16-s + 0.0731·17-s + ⋯

Functional equation

Λ(s)=(8s/2 ΓC(s) L(s)=((−0.711+0.702i) Λ‾(22−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(22-s) \end{aligned}
Λ(s)=(8s/2 ΓC(s+21/2) L(s)=((−0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 88    =    232^{3}
Sign: −0.711+0.702i-0.711 + 0.702i
Analytic conductor: 22.358122.3581
Root analytic conductor: 4.728444.72844
Motivic weight: 2121
Rational: no
Arithmetic: yes
Character: χ8(5,⋅)\chi_{8} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 8, ( :21/2), −0.711+0.702i)(2,\ 8,\ (\ :21/2),\ -0.711 + 0.702i)

Particular Values

L(11)L(11) ≈\approx 0.534129−1.30144i0.534129 - 1.30144i
L(12)L(\frac12) ≈\approx 0.534129−1.30144i0.534129 - 1.30144i
L(232)L(\frac{23}{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+(377.−1.39e3i)T 1 + (377. - 1.39e3i)T
good3 1+1.48e5iT−1.04e10T2 1 + 1.48e5iT - 1.04e10T^{2}
5 1+2.51e7iT−4.76e14T2 1 + 2.51e7iT - 4.76e14T^{2}
7 1−8.96e8T+5.58e17T2 1 - 8.96e8T + 5.58e17T^{2}
11 1+1.46e11iT−7.40e21T2 1 + 1.46e11iT - 7.40e21T^{2}
13 1−4.53e11iT−2.47e23T2 1 - 4.53e11iT - 2.47e23T^{2}
17 1−6.08e11T+6.90e25T2 1 - 6.08e11T + 6.90e25T^{2}
19 1+3.33e13iT−7.14e26T2 1 + 3.33e13iT - 7.14e26T^{2}
23 1+2.43e14T+3.94e28T2 1 + 2.43e14T + 3.94e28T^{2}
29 1−1.19e15iT−5.13e30T2 1 - 1.19e15iT - 5.13e30T^{2}
31 1−5.18e15T+2.08e31T2 1 - 5.18e15T + 2.08e31T^{2}
37 1−2.87e15iT−8.55e32T2 1 - 2.87e15iT - 8.55e32T^{2}
41 1+9.95e16T+7.38e33T2 1 + 9.95e16T + 7.38e33T^{2}
43 1−2.13e17iT−2.00e34T2 1 - 2.13e17iT - 2.00e34T^{2}
47 1+2.65e17T+1.30e35T2 1 + 2.65e17T + 1.30e35T^{2}
53 1+1.06e17iT−1.62e36T2 1 + 1.06e17iT - 1.62e36T^{2}
59 1−3.03e18iT−1.54e37T2 1 - 3.03e18iT - 1.54e37T^{2}
61 1+3.77e18iT−3.10e37T2 1 + 3.77e18iT - 3.10e37T^{2}
67 1−7.43e18iT−2.22e38T2 1 - 7.43e18iT - 2.22e38T^{2}
71 1−3.77e19T+7.52e38T2 1 - 3.77e19T + 7.52e38T^{2}
73 1+3.15e19T+1.34e39T2 1 + 3.15e19T + 1.34e39T^{2}
79 1−8.13e19T+7.08e39T2 1 - 8.13e19T + 7.08e39T^{2}
83 1+1.42e20iT−1.99e40T2 1 + 1.42e20iT - 1.99e40T^{2}
89 1−4.63e20T+8.65e40T2 1 - 4.63e20T + 8.65e40T^{2}
97 1−8.42e20T+5.27e41T2 1 - 8.42e20T + 5.27e41T^{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

−16.23353281107151424432216336230, −14.12598163215184906543001957828, −13.28840660758551046183601393823, −11.61537604353753477951574648885, −8.731227500130108269458442342233, −7.992387204579399978058872651912, −6.35341574420010375996231469120, −4.84682875667856006004387443881, −1.48704853612403231827358110360, −0.57737106757040686956272165596, 2.06784220137434101707282660143, 3.69627357905105783200711277833, 4.91203877597456706995979696593, 7.950301912108482844872567295056, 9.977248999933452415735374381166, 10.49293515509575645531244309965, 11.93398903310409163070593766942, 14.34948756943896720428099931952, 15.24744907261667675131537168487, 17.35053626553846005425032150730

Graph of the ZZ-function along the critical line