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Properties

Label 2-12e2-3.2-c2-0-1
Degree 22
Conductor 144144
Sign 0.577−0.816i0.577 - 0.816i
Analytic cond. 3.923713.92371
Root an. cond. 1.980831.98083
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.24i·5-s + 4·7-s + 16.9i·11-s + 8·13-s + 12.7i·17-s + 16·19-s − 16.9i·23-s + 7.00·25-s − 4.24i·29-s − 44·31-s + 16.9i·35-s − 34·37-s − 46.6i·41-s + 40·43-s − 84.8i·47-s + ⋯
L(s)  = 1  + 0.848i·5-s + 0.571·7-s + 1.54i·11-s + 0.615·13-s + 0.748i·17-s + 0.842·19-s − 0.737i·23-s + 0.280·25-s − 0.146i·29-s − 1.41·31-s + 0.484i·35-s − 0.918·37-s − 1.13i·41-s + 0.930·43-s − 1.80i·47-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((0.577−0.816i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1) L(s)=((0.577−0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: 0.577−0.816i0.577 - 0.816i
Analytic conductor: 3.923713.92371
Root analytic conductor: 1.980831.98083
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ144(17,⋅)\chi_{144} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :1), 0.577−0.816i)(2,\ 144,\ (\ :1),\ 0.577 - 0.816i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.31967+0.683116i1.31967 + 0.683116i
L(12)L(\frac12) ≈\approx 1.31967+0.683116i1.31967 + 0.683116i
L(2)L(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 1−4.24iT−25T2 1 - 4.24iT - 25T^{2}
7 1−4T+49T2 1 - 4T + 49T^{2}
11 1−16.9iT−121T2 1 - 16.9iT - 121T^{2}
13 1−8T+169T2 1 - 8T + 169T^{2}
17 1−12.7iT−289T2 1 - 12.7iT - 289T^{2}
19 1−16T+361T2 1 - 16T + 361T^{2}
23 1+16.9iT−529T2 1 + 16.9iT - 529T^{2}
29 1+4.24iT−841T2 1 + 4.24iT - 841T^{2}
31 1+44T+961T2 1 + 44T + 961T^{2}
37 1+34T+1.36e3T2 1 + 34T + 1.36e3T^{2}
41 1+46.6iT−1.68e3T2 1 + 46.6iT - 1.68e3T^{2}
43 1−40T+1.84e3T2 1 - 40T + 1.84e3T^{2}
47 1+84.8iT−2.20e3T2 1 + 84.8iT - 2.20e3T^{2}
53 1+38.1iT−2.80e3T2 1 + 38.1iT - 2.80e3T^{2}
59 1−33.9iT−3.48e3T2 1 - 33.9iT - 3.48e3T^{2}
61 1−50T+3.72e3T2 1 - 50T + 3.72e3T^{2}
67 1+8T+4.48e3T2 1 + 8T + 4.48e3T^{2}
71 1+50.9iT−5.04e3T2 1 + 50.9iT - 5.04e3T^{2}
73 1+16T+5.32e3T2 1 + 16T + 5.32e3T^{2}
79 1−76T+6.24e3T2 1 - 76T + 6.24e3T^{2}
83 1−118.iT−6.88e3T2 1 - 118. iT - 6.88e3T^{2}
89 1+12.7iT−7.92e3T2 1 + 12.7iT - 7.92e3T^{2}
97 1−176T+9.40e3T2 1 - 176T + 9.40e3T^{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

−12.95144612329030438916692540454, −11.99285281882888634036664730608, −10.87886488031190005676882756420, −10.14994928887121089115636409590, −8.859447563945669516645463318685, −7.56127038151087732071704044096, −6.70842282145024197873733413351, −5.21592030494207403613256592313, −3.77186660978585751289383881603, −2.01410154362396858144121923750, 1.11813309660071895897224950303, 3.35682349327906350185366723474, 4.94699627927993528046479845951, 5.92088429781282570169664953717, 7.56535336701896195563304015038, 8.602559935029808092750102300036, 9.366793536857037910290904970593, 10.95036025816715436989989699316, 11.55734411058502514096370369368, 12.78812196102597744355154839078

Graph of the ZZ-function along the critical line