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Properties

Label 2-3e5-27.16-c1-0-6
Degree 22
Conductor 243243
Sign 0.334+0.942i0.334 + 0.942i
Analytic cond. 1.940361.94036
Root an. cond. 1.392961.39296
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.614 + 0.515i)2-s + (−0.235 − 1.33i)4-s + (−2.58 − 0.940i)5-s + (0.412 − 2.34i)7-s + (1.34 − 2.33i)8-s + (−1.10 − 1.90i)10-s + (−0.235 + 0.0855i)11-s + (2.00 − 1.67i)13-s + (1.45 − 1.22i)14-s + (−0.524 + 0.190i)16-s + (−0.146 − 0.254i)17-s + (1.39 − 2.41i)19-s + (−0.648 + 3.67i)20-s + (−0.188 − 0.0685i)22-s + (1.16 + 6.58i)23-s + ⋯
L(s)  = 1  + (0.434 + 0.364i)2-s + (−0.117 − 0.668i)4-s + (−1.15 − 0.420i)5-s + (0.155 − 0.884i)7-s + (0.475 − 0.823i)8-s + (−0.348 − 0.603i)10-s + (−0.0708 + 0.0257i)11-s + (0.554 − 0.465i)13-s + (0.390 − 0.327i)14-s + (−0.131 + 0.0477i)16-s + (−0.0355 − 0.0616i)17-s + (0.319 − 0.553i)19-s + (−0.144 + 0.822i)20-s + (−0.0401 − 0.0146i)22-s + (0.242 + 1.37i)23-s + ⋯

Functional equation

Λ(s)=(243s/2 ΓC(s) L(s)=((0.334+0.942i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(243s/2 ΓC(s+1/2) L(s)=((0.334+0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 243243    =    353^{5}
Sign: 0.334+0.942i0.334 + 0.942i
Analytic conductor: 1.940361.94036
Root analytic conductor: 1.392961.39296
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ243(136,⋅)\chi_{243} (136, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 243, ( :1/2), 0.334+0.942i)(2,\ 243,\ (\ :1/2),\ 0.334 + 0.942i)

Particular Values

L(1)L(1) ≈\approx 0.991236−0.699828i0.991236 - 0.699828i
L(12)L(\frac12) ≈\approx 0.991236−0.699828i0.991236 - 0.699828i
L(32)L(\frac{3}{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)
bad3 1 1
good2 1+(−0.614−0.515i)T+(0.347+1.96i)T2 1 + (-0.614 - 0.515i)T + (0.347 + 1.96i)T^{2}
5 1+(2.58+0.940i)T+(3.83+3.21i)T2 1 + (2.58 + 0.940i)T + (3.83 + 3.21i)T^{2}
7 1+(−0.412+2.34i)T+(−6.57−2.39i)T2 1 + (-0.412 + 2.34i)T + (-6.57 - 2.39i)T^{2}
11 1+(0.235−0.0855i)T+(8.42−7.07i)T2 1 + (0.235 - 0.0855i)T + (8.42 - 7.07i)T^{2}
13 1+(−2.00+1.67i)T+(2.25−12.8i)T2 1 + (-2.00 + 1.67i)T + (2.25 - 12.8i)T^{2}
17 1+(0.146+0.254i)T+(−8.5+14.7i)T2 1 + (0.146 + 0.254i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.39+2.41i)T+(−9.5−16.4i)T2 1 + (-1.39 + 2.41i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.16−6.58i)T+(−21.6+7.86i)T2 1 + (-1.16 - 6.58i)T + (-21.6 + 7.86i)T^{2}
29 1+(−0.271−0.228i)T+(5.03+28.5i)T2 1 + (-0.271 - 0.228i)T + (5.03 + 28.5i)T^{2}
31 1+(−0.480−2.72i)T+(−29.1+10.6i)T2 1 + (-0.480 - 2.72i)T + (-29.1 + 10.6i)T^{2}
37 1+(−3.49−6.05i)T+(−18.5+32.0i)T2 1 + (-3.49 - 6.05i)T + (-18.5 + 32.0i)T^{2}
41 1+(−7.44+6.24i)T+(7.11−40.3i)T2 1 + (-7.44 + 6.24i)T + (7.11 - 40.3i)T^{2}
43 1+(0.244−0.0891i)T+(32.9−27.6i)T2 1 + (0.244 - 0.0891i)T + (32.9 - 27.6i)T^{2}
47 1+(−1.98+11.2i)T+(−44.1−16.0i)T2 1 + (-1.98 + 11.2i)T + (-44.1 - 16.0i)T^{2}
53 1+5.43T+53T2 1 + 5.43T + 53T^{2}
59 1+(−5.61−2.04i)T+(45.1+37.9i)T2 1 + (-5.61 - 2.04i)T + (45.1 + 37.9i)T^{2}
61 1+(2.05−11.6i)T+(−57.3−20.8i)T2 1 + (2.05 - 11.6i)T + (-57.3 - 20.8i)T^{2}
67 1+(−1.38+1.16i)T+(11.6−65.9i)T2 1 + (-1.38 + 1.16i)T + (11.6 - 65.9i)T^{2}
71 1+(0.185+0.320i)T+(−35.5+61.4i)T2 1 + (0.185 + 0.320i)T + (-35.5 + 61.4i)T^{2}
73 1+(2.51−4.35i)T+(−36.5−63.2i)T2 1 + (2.51 - 4.35i)T + (-36.5 - 63.2i)T^{2}
79 1+(−0.614−0.516i)T+(13.7+77.7i)T2 1 + (-0.614 - 0.516i)T + (13.7 + 77.7i)T^{2}
83 1+(−2.11−1.77i)T+(14.4+81.7i)T2 1 + (-2.11 - 1.77i)T + (14.4 + 81.7i)T^{2}
89 1+(−5.22+9.05i)T+(−44.5−77.0i)T2 1 + (-5.22 + 9.05i)T + (-44.5 - 77.0i)T^{2}
97 1+(−13.9+5.07i)T+(74.3−62.3i)T2 1 + (-13.9 + 5.07i)T + (74.3 - 62.3i)T^{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

−11.87977939278536059869681914602, −11.00709451285926415698059333703, −10.13433498266989367178984732028, −8.931615354144748414358797308164, −7.73783693639561184034104661546, −6.99922479368061964491743494678, −5.60835540233962332377433746898, −4.54849542664153722510807865347, −3.64108175830880532467656655183, −0.930147933845549804229175208277, 2.53354513126967766156119519342, 3.70326516729052356650578668524, 4.65423568644921511740676293150, 6.17945872986257115111090914139, 7.57727878032114530527210880033, 8.245018518645119213245463473597, 9.229781392161143327439377342973, 10.89688064997335796418235702953, 11.44557364329854064145017543180, 12.26664675898722879723125588317

Graph of the ZZ-function along the critical line