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Properties

Label 2-1080-8.5-c1-0-4
Degree $2$
Conductor $1080$
Sign $-0.818 + 0.574i$
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  + (−0.444 + 1.34i)2-s + (−1.60 − 1.19i)4-s + i·5-s − 1.61·7-s + (2.31 − 1.62i)8-s + (−1.34 − 0.444i)10-s − 2.72i·11-s + 6.12i·13-s + (0.718 − 2.16i)14-s + (1.15 + 3.83i)16-s + 4.31·17-s + 2.87i·19-s + (1.19 − 1.60i)20-s + (3.65 + 1.21i)22-s − 7.44·23-s + ⋯
L(s)  = 1  + (−0.314 + 0.949i)2-s + (−0.802 − 0.596i)4-s + 0.447i·5-s − 0.610·7-s + (0.818 − 0.574i)8-s + (−0.424 − 0.140i)10-s − 0.821i·11-s + 1.70i·13-s + (0.192 − 0.579i)14-s + (0.287 + 0.957i)16-s + 1.04·17-s + 0.660i·19-s + (0.266 − 0.358i)20-s + (0.779 + 0.258i)22-s − 1.55·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.818 + 0.574i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.818 + 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3911235412\)
\(L(\frac12)\) \(\approx\) \(0.3911235412\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.444 - 1.34i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 + 2.72iT - 11T^{2} \)
13 \( 1 - 6.12iT - 13T^{2} \)
17 \( 1 - 4.31T + 17T^{2} \)
19 \( 1 - 2.87iT - 19T^{2} \)
23 \( 1 + 7.44T + 23T^{2} \)
29 \( 1 + 1.43iT - 29T^{2} \)
31 \( 1 + 9.17T + 31T^{2} \)
37 \( 1 + 9.26iT - 37T^{2} \)
41 \( 1 + 6.77T + 41T^{2} \)
43 \( 1 - 7.56iT - 43T^{2} \)
47 \( 1 - 0.642T + 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 14.5iT - 59T^{2} \)
61 \( 1 - 9.00iT - 61T^{2} \)
67 \( 1 - 6.55iT - 67T^{2} \)
71 \( 1 + 3.13T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 1.53T + 79T^{2} \)
83 \( 1 + 8.44iT - 83T^{2} \)
89 \( 1 + 7.56T + 89T^{2} \)
97 \( 1 + 1.11T + 97T^{2} \)
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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

−10.05483211173457612890477598022, −9.490453250646842247187522832454, −8.695218368142716013001372621223, −7.77042101935244104642535785252, −7.05268501787687500059807592110, −6.13018726934411756621309470783, −5.69183235591328202354592754789, −4.25456500306481087354933431126, −3.48477221838247008481393798529, −1.73695221072778335997116732780, 0.19355587932377680281059335118, 1.67375539712374949087347265798, 2.98071514499721206978542227610, 3.76190445886785016012604920957, 4.96228687242651163776608950814, 5.69547000911964260076473357979, 7.13377998997142832236998401929, 7.966854291320270699190547161044, 8.633951108447839623732672539196, 9.792493892535220810771750784661

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