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Properties

Label 2-273-13.4-c1-0-14
Degree 22
Conductor 273273
Sign 0.631+0.775i0.631 + 0.775i
Analytic cond. 2.179912.17991
Root an. cond. 1.476451.47645
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  + (0.924 + 0.533i)2-s + (0.5 − 0.866i)3-s + (−0.430 − 0.745i)4-s − 0.994i·5-s + (0.924 − 0.533i)6-s + (0.866 − 0.5i)7-s − 3.05i·8-s + (−0.499 − 0.866i)9-s + (0.530 − 0.919i)10-s + (−0.215 − 0.124i)11-s − 0.860·12-s + (−3.56 + 0.529i)13-s + 1.06·14-s + (−0.860 − 0.497i)15-s + (0.769 − 1.33i)16-s + (3.10 + 5.37i)17-s + ⋯
L(s)  = 1  + (0.653 + 0.377i)2-s + (0.288 − 0.499i)3-s + (−0.215 − 0.372i)4-s − 0.444i·5-s + (0.377 − 0.217i)6-s + (0.327 − 0.188i)7-s − 1.07i·8-s + (−0.166 − 0.288i)9-s + (0.167 − 0.290i)10-s + (−0.0650 − 0.0375i)11-s − 0.248·12-s + (−0.989 + 0.146i)13-s + 0.285·14-s + (−0.222 − 0.128i)15-s + (0.192 − 0.333i)16-s + (0.752 + 1.30i)17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.631+0.775i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.631+0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.631+0.775i0.631 + 0.775i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(43,⋅)\chi_{273} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.631+0.775i)(2,\ 273,\ (\ :1/2),\ 0.631 + 0.775i)

Particular Values

L(1)L(1) ≈\approx 1.61031−0.764887i1.61031 - 0.764887i
L(12)L(\frac12) ≈\approx 1.61031−0.764887i1.61031 - 0.764887i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
13 1+(3.56−0.529i)T 1 + (3.56 - 0.529i)T
good2 1+(−0.924−0.533i)T+(1+1.73i)T2 1 + (-0.924 - 0.533i)T + (1 + 1.73i)T^{2}
5 1+0.994iT−5T2 1 + 0.994iT - 5T^{2}
11 1+(0.215+0.124i)T+(5.5+9.52i)T2 1 + (0.215 + 0.124i)T + (5.5 + 9.52i)T^{2}
17 1+(−3.10−5.37i)T+(−8.5+14.7i)T2 1 + (-3.10 - 5.37i)T + (-8.5 + 14.7i)T^{2}
19 1+(−7.28+4.20i)T+(9.5−16.4i)T2 1 + (-7.28 + 4.20i)T + (9.5 - 16.4i)T^{2}
23 1+(0.907−1.57i)T+(−11.5−19.9i)T2 1 + (0.907 - 1.57i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.34−2.32i)T+(−14.5−25.1i)T2 1 + (1.34 - 2.32i)T + (-14.5 - 25.1i)T^{2}
31 1−8.53iT−31T2 1 - 8.53iT - 31T^{2}
37 1+(3.66+2.11i)T+(18.5+32.0i)T2 1 + (3.66 + 2.11i)T + (18.5 + 32.0i)T^{2}
41 1+(−5.57−3.21i)T+(20.5+35.5i)T2 1 + (-5.57 - 3.21i)T + (20.5 + 35.5i)T^{2}
43 1+(5.71+9.89i)T+(−21.5+37.2i)T2 1 + (5.71 + 9.89i)T + (-21.5 + 37.2i)T^{2}
47 1−10.6iT−47T2 1 - 10.6iT - 47T^{2}
53 1+0.601T+53T2 1 + 0.601T + 53T^{2}
59 1+(7.71−4.45i)T+(29.5−51.0i)T2 1 + (7.71 - 4.45i)T + (29.5 - 51.0i)T^{2}
61 1+(2.87+4.97i)T+(−30.5+52.8i)T2 1 + (2.87 + 4.97i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.73−3.31i)T+(33.5+58.0i)T2 1 + (-5.73 - 3.31i)T + (33.5 + 58.0i)T^{2}
71 1+(−10.4+6.00i)T+(35.5−61.4i)T2 1 + (-10.4 + 6.00i)T + (35.5 - 61.4i)T^{2}
73 1+9.70iT−73T2 1 + 9.70iT - 73T^{2}
79 1+5.63T+79T2 1 + 5.63T + 79T^{2}
83 1−16.5iT−83T2 1 - 16.5iT - 83T^{2}
89 1+(8.47+4.89i)T+(44.5+77.0i)T2 1 + (8.47 + 4.89i)T + (44.5 + 77.0i)T^{2}
97 1+(−2.10+1.21i)T+(48.5−84.0i)T2 1 + (-2.10 + 1.21i)T + (48.5 - 84.0i)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

−12.24456115666942736931115992896, −10.83688133239995747507969965189, −9.761033520791199629884848530722, −8.893611133017276410792124011485, −7.65194647095589810806352531479, −6.81889175336447890753830842084, −5.51333941367982064165923186739, −4.78344183299259218750180192738, −3.33944192466494705197134553226, −1.30027106143439142890700495617, 2.55837453557275015142563718932, 3.45507190714334913256543121503, 4.76156505754206309446520176625, 5.53575342336017049397703089183, 7.37860916359488873979378671446, 8.057711228987438188039566238852, 9.387076582688803430612710291332, 10.09636955394070426595318397427, 11.44525688243233719588643798882, 11.90415115896720172561142641539

Graph of the ZZ-function along the critical line