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Properties

Label 2-270-45.13-c2-0-2
Degree 22
Conductor 270270
Sign −0.900−0.433i-0.900 - 0.433i
Analytic cond. 7.356967.35696
Root an. cond. 2.712372.71237
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (−4.54 + 2.07i)5-s + (−9.78 − 2.62i)7-s + (1.99 + 2i)8-s + (−6.97 + 1.16i)10-s + (8.11 + 14.0i)11-s + (−20.2 + 5.41i)13-s + (−12.4 − 7.16i)14-s + (1.99 + 3.46i)16-s + (−7.49 + 7.49i)17-s − 1.15i·19-s + (−9.95 − 0.954i)20-s + (5.93 + 22.1i)22-s + (−19.1 + 5.14i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.250i)4-s + (−0.909 + 0.415i)5-s + (−1.39 − 0.374i)7-s + (0.249 + 0.250i)8-s + (−0.697 + 0.116i)10-s + (0.737 + 1.27i)11-s + (−1.55 + 0.416i)13-s + (−0.886 − 0.511i)14-s + (0.124 + 0.216i)16-s + (−0.441 + 0.441i)17-s − 0.0607i·19-s + (−0.497 − 0.0477i)20-s + (0.269 + 1.00i)22-s + (−0.834 + 0.223i)23-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((−0.900−0.433i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.433i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((−0.900−0.433i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.900 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: −0.900−0.433i-0.900 - 0.433i
Analytic conductor: 7.356967.35696
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ270(253,⋅)\chi_{270} (253, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1), −0.900−0.433i)(2,\ 270,\ (\ :1),\ -0.900 - 0.433i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.189969+0.832269i0.189969 + 0.832269i
L(12)L(\frac12) ≈\approx 0.189969+0.832269i0.189969 + 0.832269i
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+(−1.36−0.366i)T 1 + (-1.36 - 0.366i)T
3 1 1
5 1+(4.54−2.07i)T 1 + (4.54 - 2.07i)T
good7 1+(9.78+2.62i)T+(42.4+24.5i)T2 1 + (9.78 + 2.62i)T + (42.4 + 24.5i)T^{2}
11 1+(−8.11−14.0i)T+(−60.5+104.i)T2 1 + (-8.11 - 14.0i)T + (-60.5 + 104. i)T^{2}
13 1+(20.2−5.41i)T+(146.−84.5i)T2 1 + (20.2 - 5.41i)T + (146. - 84.5i)T^{2}
17 1+(7.49−7.49i)T−289iT2 1 + (7.49 - 7.49i)T - 289iT^{2}
19 1+1.15iT−361T2 1 + 1.15iT - 361T^{2}
23 1+(19.1−5.14i)T+(458.−264.5i)T2 1 + (19.1 - 5.14i)T + (458. - 264.5i)T^{2}
29 1+(−19.9+11.5i)T+(420.5−728.i)T2 1 + (-19.9 + 11.5i)T + (420.5 - 728. i)T^{2}
31 1+(−2.50+4.34i)T+(−480.5−832.i)T2 1 + (-2.50 + 4.34i)T + (-480.5 - 832. i)T^{2}
37 1+(−33.0+33.0i)T−1.36e3iT2 1 + (-33.0 + 33.0i)T - 1.36e3iT^{2}
41 1+(26.2−45.4i)T+(−840.5−1.45e3i)T2 1 + (26.2 - 45.4i)T + (-840.5 - 1.45e3i)T^{2}
43 1+(8.94−33.3i)T+(−1.60e3−924.5i)T2 1 + (8.94 - 33.3i)T + (-1.60e3 - 924.5i)T^{2}
47 1+(15.5+4.17i)T+(1.91e3+1.10e3i)T2 1 + (15.5 + 4.17i)T + (1.91e3 + 1.10e3i)T^{2}
53 1+(−30.1−30.1i)T+2.80e3iT2 1 + (-30.1 - 30.1i)T + 2.80e3iT^{2}
59 1+(−0.730−0.421i)T+(1.74e3+3.01e3i)T2 1 + (-0.730 - 0.421i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(15.8+27.5i)T+(−1.86e3+3.22e3i)T2 1 + (15.8 + 27.5i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(9.02+33.6i)T+(−3.88e3+2.24e3i)T2 1 + (9.02 + 33.6i)T + (-3.88e3 + 2.24e3i)T^{2}
71 1+1.68T+5.04e3T2 1 + 1.68T + 5.04e3T^{2}
73 1+(−100.−100.i)T+5.32e3iT2 1 + (-100. - 100. i)T + 5.32e3iT^{2}
79 1+(9.01−5.20i)T+(3.12e3−5.40e3i)T2 1 + (9.01 - 5.20i)T + (3.12e3 - 5.40e3i)T^{2}
83 1+(6.02−22.4i)T+(−5.96e3−3.44e3i)T2 1 + (6.02 - 22.4i)T + (-5.96e3 - 3.44e3i)T^{2}
89 1−102.iT−7.92e3T2 1 - 102. iT - 7.92e3T^{2}
97 1+(21.8+5.85i)T+(8.14e3+4.70e3i)T2 1 + (21.8 + 5.85i)T + (8.14e3 + 4.70e3i)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

−12.28988838025930420473715626979, −11.46319102363113564281520916679, −10.12083535147547913989649454072, −9.505437305079726069315476381636, −7.85256204569371295992875881715, −6.96471599725986176357750718078, −6.41621148623826071800359046283, −4.61190825078706272500793570273, −3.86241585160708553465462308696, −2.53170403077611751253859308608, 0.32539369701680413845902081507, 2.81362877975907162664079370679, 3.74091021313876944661419780509, 5.01068231733160198636148705975, 6.20513064143419320831689284074, 7.13167491622910228103454931999, 8.429043113949942195959399961161, 9.428951519551830840070332424365, 10.43486309191310895634421877723, 11.74077573756493056084059913390

Graph of the ZZ-function along the critical line