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Properties

Label 2-1080-9.7-c1-0-4
Degree 22
Conductor 10801080
Sign 0.642−0.766i0.642 - 0.766i
Analytic cond. 8.623848.62384
Root an. cond. 2.936632.93663
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  + (−0.5 − 0.866i)5-s + (−0.133 + 0.232i)7-s + (−0.732 + 1.26i)11-s + (2.73 + 4.73i)13-s − 0.535·17-s − 2·19-s + (1.86 + 3.23i)23-s + (−0.499 + 0.866i)25-s + (0.767 − 1.33i)29-s + (1 + 1.73i)31-s + 0.267·35-s + 10.3·37-s + (4.96 + 8.59i)41-s + (2.26 − 3.92i)43-s + (0.133 − 0.232i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.0506 + 0.0877i)7-s + (−0.220 + 0.382i)11-s + (0.757 + 1.31i)13-s − 0.129·17-s − 0.458·19-s + (0.389 + 0.673i)23-s + (−0.0999 + 0.173i)25-s + (0.142 − 0.246i)29-s + (0.179 + 0.311i)31-s + 0.0452·35-s + 1.70·37-s + (0.775 + 1.34i)41-s + (0.345 − 0.599i)43-s + (0.0195 − 0.0338i)47-s + ⋯

Functional equation

Λ(s)=(1080s/2 ΓC(s) L(s)=((0.642−0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1080s/2 ΓC(s+1/2) L(s)=((0.642−0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10801080    =    23⋅33⋅52^{3} \cdot 3^{3} \cdot 5
Sign: 0.642−0.766i0.642 - 0.766i
Analytic conductor: 8.623848.62384
Root analytic conductor: 2.936632.93663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1080(721,⋅)\chi_{1080} (721, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1080, ( :1/2), 0.642−0.766i)(2,\ 1080,\ (\ :1/2),\ 0.642 - 0.766i)

Particular Values

L(1)L(1) ≈\approx 1.4208732801.420873280
L(12)L(\frac12) ≈\approx 1.4208732801.420873280
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)
bad2 1 1
3 1 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(0.133−0.232i)T+(−3.5−6.06i)T2 1 + (0.133 - 0.232i)T + (-3.5 - 6.06i)T^{2}
11 1+(0.732−1.26i)T+(−5.5−9.52i)T2 1 + (0.732 - 1.26i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.73−4.73i)T+(−6.5+11.2i)T2 1 + (-2.73 - 4.73i)T + (-6.5 + 11.2i)T^{2}
17 1+0.535T+17T2 1 + 0.535T + 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 1+(−1.86−3.23i)T+(−11.5+19.9i)T2 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.767+1.33i)T+(−14.5−25.1i)T2 1 + (-0.767 + 1.33i)T + (-14.5 - 25.1i)T^{2}
31 1+(−1−1.73i)T+(−15.5+26.8i)T2 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1−10.3T+37T2 1 - 10.3T + 37T^{2}
41 1+(−4.96−8.59i)T+(−20.5+35.5i)T2 1 + (-4.96 - 8.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.26+3.92i)T+(−21.5−37.2i)T2 1 + (-2.26 + 3.92i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.133+0.232i)T+(−23.5−40.7i)T2 1 + (-0.133 + 0.232i)T + (-23.5 - 40.7i)T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+(−7.19−12.4i)T+(−29.5+51.0i)T2 1 + (-7.19 - 12.4i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.23+7.33i)T+(−30.5−52.8i)T2 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.13+5.42i)T+(−33.5+58.0i)T2 1 + (3.13 + 5.42i)T + (-33.5 + 58.0i)T^{2}
71 1+9.46T+71T2 1 + 9.46T + 71T^{2}
73 1−6.92T+73T2 1 - 6.92T + 73T^{2}
79 1+(7.73−13.3i)T+(−39.5−68.4i)T2 1 + (7.73 - 13.3i)T + (-39.5 - 68.4i)T^{2}
83 1+(6.59−11.4i)T+(−41.5−71.8i)T2 1 + (6.59 - 11.4i)T + (-41.5 - 71.8i)T^{2}
89 1−9.92T+89T2 1 - 9.92T + 89T^{2}
97 1+(−4.46+7.73i)T+(−48.5−84.0i)T2 1 + (-4.46 + 7.73i)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

−9.806931992677797036668577188189, −9.190980676044023226876661899040, −8.408966440346788438269774490046, −7.53774525947224234439202795260, −6.61614939482827803528372994325, −5.79887547590887106877400891688, −4.61393647928144339991092765374, −3.99304315006105007549912713048, −2.60941583153541078735310266873, −1.31124414336993506038135382249, 0.71125349592176876464873192494, 2.49723374115607971705597028005, 3.42157947195890448104651927741, 4.44027596240149613295539198232, 5.63035085610172494694306802839, 6.30369870221744823030939251627, 7.34229469200416557172322559908, 8.138216893104451471742300746311, 8.797824898149567238167237228235, 9.892501823768718525558132072191

Graph of the ZZ-function along the critical line