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Properties

Label 2-270-135.103-c2-0-19
Degree 22
Conductor 270270
Sign 0.0452+0.998i0.0452 + 0.998i
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  + (0.123 − 1.40i)2-s + (−2.99 + 0.103i)3-s + (−1.96 − 0.347i)4-s + (3.21 − 3.83i)5-s + (−0.223 + 4.23i)6-s + (7.14 + 5.00i)7-s + (−0.732 + 2.73i)8-s + (8.97 − 0.620i)9-s + (−5.00 − 4.99i)10-s + (−0.513 − 0.186i)11-s + (5.94 + 0.837i)12-s + (0.585 + 6.69i)13-s + (7.93 − 9.45i)14-s + (−9.22 + 11.8i)15-s + (3.75 + 1.36i)16-s + (−4.16 − 15.5i)17-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (−0.999 + 0.0344i)3-s + (−0.492 − 0.0868i)4-s + (0.642 − 0.766i)5-s + (−0.0372 + 0.706i)6-s + (1.02 + 0.715i)7-s + (−0.0915 + 0.341i)8-s + (0.997 − 0.0689i)9-s + (−0.500 − 0.499i)10-s + (−0.0466 − 0.0169i)11-s + (0.495 + 0.0697i)12-s + (0.0450 + 0.515i)13-s + (0.566 − 0.675i)14-s + (−0.615 + 0.788i)15-s + (0.234 + 0.0855i)16-s + (−0.245 − 0.914i)17-s + ⋯

Functional equation

Λ(s)=(270s/2 ΓC(s) L(s)=((0.0452+0.998i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0452 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(270s/2 ΓC(s+1) L(s)=((0.0452+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0452 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 270270    =    2⋅33⋅52 \cdot 3^{3} \cdot 5
Sign: 0.0452+0.998i0.0452 + 0.998i
Analytic conductor: 7.356967.35696
Root analytic conductor: 2.712372.71237
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ270(103,⋅)\chi_{270} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 270, ( :1), 0.0452+0.998i)(2,\ 270,\ (\ :1),\ 0.0452 + 0.998i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.01518−0.970202i1.01518 - 0.970202i
L(12)L(\frac12) ≈\approx 1.01518−0.970202i1.01518 - 0.970202i
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+(−0.123+1.40i)T 1 + (-0.123 + 1.40i)T
3 1+(2.99−0.103i)T 1 + (2.99 - 0.103i)T
5 1+(−3.21+3.83i)T 1 + (-3.21 + 3.83i)T
good7 1+(−7.14−5.00i)T+(16.7+46.0i)T2 1 + (-7.14 - 5.00i)T + (16.7 + 46.0i)T^{2}
11 1+(0.513+0.186i)T+(92.6+77.7i)T2 1 + (0.513 + 0.186i)T + (92.6 + 77.7i)T^{2}
13 1+(−0.585−6.69i)T+(−166.+29.3i)T2 1 + (-0.585 - 6.69i)T + (-166. + 29.3i)T^{2}
17 1+(4.16+15.5i)T+(−250.+144.5i)T2 1 + (4.16 + 15.5i)T + (-250. + 144.5i)T^{2}
19 1+(−16.6+9.60i)T+(180.5−312.i)T2 1 + (-16.6 + 9.60i)T + (180.5 - 312. i)T^{2}
23 1+(−28.2+19.7i)T+(180.−497.i)T2 1 + (-28.2 + 19.7i)T + (180. - 497. i)T^{2}
29 1+(31.8+37.9i)T+(−146.+828.i)T2 1 + (31.8 + 37.9i)T + (-146. + 828. i)T^{2}
31 1+(−1.03+5.89i)T+(−903.−328.i)T2 1 + (-1.03 + 5.89i)T + (-903. - 328. i)T^{2}
37 1+(0.836+3.12i)T+(−1.18e3+684.5i)T2 1 + (0.836 + 3.12i)T + (-1.18e3 + 684.5i)T^{2}
41 1+(−2.89−2.43i)T+(291.+1.65e3i)T2 1 + (-2.89 - 2.43i)T + (291. + 1.65e3i)T^{2}
43 1+(−22.5−48.4i)T+(−1.18e3+1.41e3i)T2 1 + (-22.5 - 48.4i)T + (-1.18e3 + 1.41e3i)T^{2}
47 1+(−50.4−35.3i)T+(755.+2.07e3i)T2 1 + (-50.4 - 35.3i)T + (755. + 2.07e3i)T^{2}
53 1+(−54.4−54.4i)T+2.80e3iT2 1 + (-54.4 - 54.4i)T + 2.80e3iT^{2}
59 1+(11.8+32.6i)T+(−2.66e3+2.23e3i)T2 1 + (11.8 + 32.6i)T + (-2.66e3 + 2.23e3i)T^{2}
61 1+(14.9+84.9i)T+(−3.49e3+1.27e3i)T2 1 + (14.9 + 84.9i)T + (-3.49e3 + 1.27e3i)T^{2}
67 1+(124.−10.9i)T+(4.42e3−779.i)T2 1 + (124. - 10.9i)T + (4.42e3 - 779. i)T^{2}
71 1+(−29.1+50.4i)T+(−2.52e3−4.36e3i)T2 1 + (-29.1 + 50.4i)T + (-2.52e3 - 4.36e3i)T^{2}
73 1+(14.1−52.8i)T+(−4.61e3−2.66e3i)T2 1 + (14.1 - 52.8i)T + (-4.61e3 - 2.66e3i)T^{2}
79 1+(−15.4−18.4i)T+(−1.08e3+6.14e3i)T2 1 + (-15.4 - 18.4i)T + (-1.08e3 + 6.14e3i)T^{2}
83 1+(−86.8−7.59i)T+(6.78e3+1.19e3i)T2 1 + (-86.8 - 7.59i)T + (6.78e3 + 1.19e3i)T^{2}
89 1+(−79.5+45.9i)T+(3.96e3−6.85e3i)T2 1 + (-79.5 + 45.9i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(23.6−11.0i)T+(6.04e3−7.20e3i)T2 1 + (23.6 - 11.0i)T + (6.04e3 - 7.20e3i)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

−11.52026061514542328686625512794, −10.82834222922836696413653690600, −9.522801058577310776313704442427, −9.012212075096953288087745903571, −7.59241368348300733913217282821, −6.07912835558502715520031086725, −5.10381043030278892870219139709, −4.52090167167503493470172400783, −2.30404231811052871928384728777, −0.949667108732143733323325108110, 1.40864474362648659985355707682, 3.73473049826261458690350040752, 5.16430072019476749951644740500, 5.78766530846223749212321076846, 7.08603812159504954997187131167, 7.53768780913811998392900324114, 9.051897597997219927063695564732, 10.38707751404934198828302421743, 10.75207114068298669124773989876, 11.81132643920846809966696505201

Graph of the ZZ-function along the critical line