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Properties

Label 2-1080-9.7-c1-0-4
Degree $2$
Conductor $1080$
Sign $0.642 - 0.766i$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420873280\)
\(L(\frac12)\) \(\approx\) \(1.420873280\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line