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Properties

Label 2-270-135.104-c2-0-16
Degree 22
Conductor 270270
Sign 0.359+0.933i0.359 + 0.933i
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  + (−1.32 − 0.483i)2-s + (−1.82 + 2.38i)3-s + (1.53 + 1.28i)4-s + (3.82 + 3.22i)5-s + (3.57 − 2.28i)6-s + (−7.39 − 8.81i)7-s + (−1.41 − 2.44i)8-s + (−2.36 − 8.68i)9-s + (−3.51 − 6.13i)10-s + (−7.46 + 1.31i)11-s + (−5.85 + 1.31i)12-s + (3.17 + 8.73i)13-s + (5.56 + 15.2i)14-s + (−14.6 + 3.24i)15-s + (0.694 + 3.93i)16-s + (15.9 − 27.5i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.607 + 0.794i)3-s + (0.383 + 0.321i)4-s + (0.764 + 0.644i)5-s + (0.595 − 0.381i)6-s + (−1.05 − 1.25i)7-s + (−0.176 − 0.306i)8-s + (−0.262 − 0.964i)9-s + (−0.351 − 0.613i)10-s + (−0.678 + 0.119i)11-s + (−0.487 + 0.109i)12-s + (0.244 + 0.671i)13-s + (0.397 + 1.09i)14-s + (−0.976 + 0.216i)15-s + (0.0434 + 0.246i)16-s + (0.936 − 1.62i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.359+0.933i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((0.359+0.933i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.359+0.933i0.359 + 0.933i
Analytic conductor: 7.356967.35696
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ270(239,⋅)\chi_{270} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1), 0.359+0.933i)(2,\ 270,\ (\ :1),\ 0.359 + 0.933i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.552874−0.379587i0.552874 - 0.379587i
L(12)L(\frac12) ≈\approx 0.552874−0.379587i0.552874 - 0.379587i
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+(1.32+0.483i)T 1 + (1.32 + 0.483i)T
3 1+(1.82−2.38i)T 1 + (1.82 - 2.38i)T
5 1+(−3.82−3.22i)T 1 + (-3.82 - 3.22i)T
good7 1+(7.39+8.81i)T+(−8.50+48.2i)T2 1 + (7.39 + 8.81i)T + (-8.50 + 48.2i)T^{2}
11 1+(7.46−1.31i)T+(113.−41.3i)T2 1 + (7.46 - 1.31i)T + (113. - 41.3i)T^{2}
13 1+(−3.17−8.73i)T+(−129.+108.i)T2 1 + (-3.17 - 8.73i)T + (-129. + 108. i)T^{2}
17 1+(−15.9+27.5i)T+(−144.5−250.i)T2 1 + (-15.9 + 27.5i)T + (-144.5 - 250. i)T^{2}
19 1+(3.87+6.71i)T+(−180.5+312.i)T2 1 + (3.87 + 6.71i)T + (-180.5 + 312. i)T^{2}
23 1+(−4.47−3.75i)T+(91.8+520.i)T2 1 + (-4.47 - 3.75i)T + (91.8 + 520. i)T^{2}
29 1+(−18.6+51.2i)T+(−644.−540.i)T2 1 + (-18.6 + 51.2i)T + (-644. - 540. i)T^{2}
31 1+(9.04+7.59i)T+(166.+946.i)T2 1 + (9.04 + 7.59i)T + (166. + 946. i)T^{2}
37 1+(39.4+22.7i)T+(684.5+1.18e3i)T2 1 + (39.4 + 22.7i)T + (684.5 + 1.18e3i)T^{2}
41 1+(8.85+24.3i)T+(−1.28e3+1.08e3i)T2 1 + (8.85 + 24.3i)T + (-1.28e3 + 1.08e3i)T^{2}
43 1+(−22.9+4.04i)T+(1.73e3−632.i)T2 1 + (-22.9 + 4.04i)T + (1.73e3 - 632. i)T^{2}
47 1+(−28.0+23.5i)T+(383.−2.17e3i)T2 1 + (-28.0 + 23.5i)T + (383. - 2.17e3i)T^{2}
53 1−99.3T+2.80e3T2 1 - 99.3T + 2.80e3T^{2}
59 1+(26.8+4.73i)T+(3.27e3+1.19e3i)T2 1 + (26.8 + 4.73i)T + (3.27e3 + 1.19e3i)T^{2}
61 1+(−14.8+12.4i)T+(646.−3.66e3i)T2 1 + (-14.8 + 12.4i)T + (646. - 3.66e3i)T^{2}
67 1+(24.4+67.0i)T+(−3.43e3+2.88e3i)T2 1 + (24.4 + 67.0i)T + (-3.43e3 + 2.88e3i)T^{2}
71 1+(−13.2−7.64i)T+(2.52e3+4.36e3i)T2 1 + (-13.2 - 7.64i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(−6.00+3.46i)T+(2.66e3−4.61e3i)T2 1 + (-6.00 + 3.46i)T + (2.66e3 - 4.61e3i)T^{2}
79 1+(1.96+0.713i)T+(4.78e3+4.01e3i)T2 1 + (1.96 + 0.713i)T + (4.78e3 + 4.01e3i)T^{2}
83 1+(125.+45.5i)T+(5.27e3+4.42e3i)T2 1 + (125. + 45.5i)T + (5.27e3 + 4.42e3i)T^{2}
89 1+(102.−59.3i)T+(3.96e3−6.85e3i)T2 1 + (102. - 59.3i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(−13.0+2.30i)T+(8.84e3−3.21e3i)T2 1 + (-13.0 + 2.30i)T + (8.84e3 - 3.21e3i)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

−11.15963658572418058992660762164, −10.33820371045528391573771514497, −9.873577117812572521381489240454, −9.157196514298198897179101224999, −7.34821640770944217130220507584, −6.69591836557559726555424941958, −5.52019028419870771843564190789, −3.96936042306767654603412446965, −2.79754811137901894770547160872, −0.47566575737690385662359758992, 1.38945355623214838107919203157, 2.80715748575277113994644135128, 5.47498703609362232669796849213, 5.78135888021477518127352198501, 6.79387411127624395606669076475, 8.252491774833794656763563710073, 8.757679259542057770799352098821, 10.07684122758919547426353002973, 10.64574941711199226159100185263, 12.28152523219411836104548742603

Graph of the ZZ-function along the critical line