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Properties

Label 2-1080-120.59-c1-0-27
Degree $2$
Conductor $1080$
Sign $-0.592 - 0.805i$
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  + (0.621 + 1.27i)2-s + (−1.22 + 1.57i)4-s + (−1.66 − 1.49i)5-s + 3.65·7-s + (−2.76 − 0.576i)8-s + (0.863 − 3.04i)10-s + 4.07i·11-s − 3.48·13-s + (2.27 + 4.64i)14-s + (−0.989 − 3.87i)16-s + 4.02·17-s + 6.53·19-s + (4.40 − 0.794i)20-s + (−5.17 + 2.53i)22-s + 7.28i·23-s + ⋯
L(s)  = 1  + (0.439 + 0.898i)2-s + (−0.613 + 0.789i)4-s + (−0.743 − 0.668i)5-s + 1.38·7-s + (−0.979 − 0.203i)8-s + (0.273 − 0.961i)10-s + 1.22i·11-s − 0.967·13-s + (0.607 + 1.24i)14-s + (−0.247 − 0.968i)16-s + 0.976·17-s + 1.50·19-s + (0.984 − 0.177i)20-s + (−1.10 + 0.540i)22-s + 1.51i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.592 - 0.805i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.592 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.664583472\)
\(L(\frac12)\) \(\approx\) \(1.664583472\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.621 - 1.27i)T \)
3 \( 1 \)
5 \( 1 + (1.66 + 1.49i)T \)
good7 \( 1 - 3.65T + 7T^{2} \)
11 \( 1 - 4.07iT - 11T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 - 4.02T + 17T^{2} \)
19 \( 1 - 6.53T + 19T^{2} \)
23 \( 1 - 7.28iT - 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 - 3.54iT - 31T^{2} \)
37 \( 1 - 4.83T + 37T^{2} \)
41 \( 1 - 6.13iT - 41T^{2} \)
43 \( 1 - 9.36iT - 43T^{2} \)
47 \( 1 + 1.86iT - 47T^{2} \)
53 \( 1 - 5.68iT - 53T^{2} \)
59 \( 1 + 7.36iT - 59T^{2} \)
61 \( 1 + 9.75iT - 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 1.33T + 71T^{2} \)
73 \( 1 - 3.96iT - 73T^{2} \)
79 \( 1 + 9.27iT - 79T^{2} \)
83 \( 1 + 5.50T + 83T^{2} \)
89 \( 1 + 1.49iT - 89T^{2} \)
97 \( 1 + 11.2iT - 97T^{2} \)
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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

−9.718502064377496852701016860463, −9.348230270981971908223333510006, −8.059170451310801715665446564454, −7.59722647662689404516093152622, −7.27296590255935188539554744797, −5.55131988635011572932982535938, −5.05053413231403535449209657943, −4.38591973928329193502172416617, −3.28967651886318316754287329418, −1.49875749519390479951861782079, 0.70070827419883125040751233866, 2.23890928576832126308617567639, 3.26103938448924475892890577777, 4.12649357129436563590267974781, 5.16083516168209230323386552434, 5.82447550245215195167850714468, 7.27508428315272485586304934367, 7.945817237728141121637530861748, 8.805938944087729414370906727189, 9.833268642173040103440478908363

Graph of the $Z$-function along the critical line