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Properties

Label 2-2e2-4.3-c18-0-0
Degree 22
Conductor 44
Sign 0.671−0.741i0.671 - 0.741i
Analytic cond. 8.215448.21544
Root an. cond. 2.866252.86625
Motivic weight 1818
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−207. − 468. i)2-s − 4.12e3i·3-s + (−1.75e5 + 1.94e5i)4-s − 1.32e6·5-s + (−1.93e6 + 8.56e5i)6-s + 1.88e7i·7-s + (1.27e8 + 4.20e7i)8-s + 3.70e8·9-s + (2.74e8 + 6.18e8i)10-s + 2.94e9i·11-s + (8.02e8 + 7.26e8i)12-s + 2.06e9·13-s + (8.84e9 − 3.92e9i)14-s + 5.45e9i·15-s + (−6.79e9 − 6.83e10i)16-s − 1.59e11·17-s + ⋯
L(s)  = 1  + (−0.405 − 0.914i)2-s − 0.209i·3-s + (−0.671 + 0.741i)4-s − 0.676·5-s + (−0.191 + 0.0850i)6-s + 0.468i·7-s + (0.949 + 0.313i)8-s + 0.956·9-s + (0.274 + 0.618i)10-s + 1.24i·11-s + (0.155 + 0.140i)12-s + 0.194·13-s + (0.427 − 0.189i)14-s + 0.141i·15-s + (−0.0988 − 0.995i)16-s − 1.34·17-s + ⋯

Functional equation

Λ(s)=(4s/2 ΓC(s) L(s)=((0.671−0.741i) Λ‾(19−s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(19-s) \end{aligned}
Λ(s)=(4s/2 ΓC(s+9) L(s)=((0.671−0.741i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44    =    222^{2}
Sign: 0.671−0.741i0.671 - 0.741i
Analytic conductor: 8.215448.21544
Root analytic conductor: 2.866252.86625
Motivic weight: 1818
Rational: no
Arithmetic: yes
Character: χ4(3,⋅)\chi_{4} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4, ( :9), 0.671−0.741i)(2,\ 4,\ (\ :9),\ 0.671 - 0.741i)

Particular Values

L(192)L(\frac{19}{2}) ≈\approx 0.736972+0.326864i0.736972 + 0.326864i
L(12)L(\frac12) ≈\approx 0.736972+0.326864i0.736972 + 0.326864i
L(10)L(10) 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+(207.+468.i)T 1 + (207. + 468. i)T
good3 1+4.12e3iT−3.87e8T2 1 + 4.12e3iT - 3.87e8T^{2}
5 1+1.32e6T+3.81e12T2 1 + 1.32e6T + 3.81e12T^{2}
7 1−1.88e7iT−1.62e15T2 1 - 1.88e7iT - 1.62e15T^{2}
11 1−2.94e9iT−5.55e18T2 1 - 2.94e9iT - 5.55e18T^{2}
13 1−2.06e9T+1.12e20T2 1 - 2.06e9T + 1.12e20T^{2}
17 1+1.59e11T+1.40e22T2 1 + 1.59e11T + 1.40e22T^{2}
19 1−3.75e11iT−1.04e23T2 1 - 3.75e11iT - 1.04e23T^{2}
23 1−3.51e12iT−3.24e24T2 1 - 3.51e12iT - 3.24e24T^{2}
29 1−1.15e12T+2.10e26T2 1 - 1.15e12T + 2.10e26T^{2}
31 1+2.94e13iT−6.99e26T2 1 + 2.94e13iT - 6.99e26T^{2}
37 1−1.43e14T+1.68e28T2 1 - 1.43e14T + 1.68e28T^{2}
41 1+3.41e14T+1.07e29T2 1 + 3.41e14T + 1.07e29T^{2}
43 1−1.16e14iT−2.52e29T2 1 - 1.16e14iT - 2.52e29T^{2}
47 1−1.28e15iT−1.25e30T2 1 - 1.28e15iT - 1.25e30T^{2}
53 1+9.01e14T+1.08e31T2 1 + 9.01e14T + 1.08e31T^{2}
59 1+1.35e16iT−7.50e31T2 1 + 1.35e16iT - 7.50e31T^{2}
61 1−1.92e15T+1.36e32T2 1 - 1.92e15T + 1.36e32T^{2}
67 1−3.28e16iT−7.40e32T2 1 - 3.28e16iT - 7.40e32T^{2}
71 1−1.05e16iT−2.10e33T2 1 - 1.05e16iT - 2.10e33T^{2}
73 1+5.27e16T+3.46e33T2 1 + 5.27e16T + 3.46e33T^{2}
79 1+2.46e16iT−1.43e34T2 1 + 2.46e16iT - 1.43e34T^{2}
83 1+1.57e17iT−3.49e34T2 1 + 1.57e17iT - 3.49e34T^{2}
89 1+1.76e17T+1.22e35T2 1 + 1.76e17T + 1.22e35T^{2}
97 1−8.50e16T+5.77e35T2 1 - 8.50e16T + 5.77e35T^{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

−20.34571198220967397692843563119, −19.00190758990131536135033455742, −17.72161780219575275169908398381, −15.54618515575670601694238318600, −13.02330194840678990713526303929, −11.66607022821518641598049061629, −9.692240712770639587396648781700, −7.65251093993026270030348420835, −4.14517795804589275651047091905, −1.79520958841218477442810051099, 0.49163872616787734541886495472, 4.37445666083121237427271949221, 6.81331802226156844595576856278, 8.626510779856763090465269347117, 10.71803566192251030818862201420, 13.49216942491982570938235538336, 15.39421368100598034060522978462, 16.48525854322644381185407175302, 18.32617090184924152968053851625, 19.74051055015451642268897876038

Graph of the ZZ-function along the critical line