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Properties

Label 2-1080-9.4-c1-0-8
Degree $2$
Conductor $1080$
Sign $0.173 + 0.984i$
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 + (−2.5 − 4.33i)11-s − 3·17-s + 5·19-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + (−5 − 8.66i)29-s + (1 − 1.73i)31-s + 4·37-s + (−1.5 + 2.59i)41-s + (−1.5 − 2.59i)43-s + (2 + 3.46i)47-s + (3.5 − 6.06i)49-s + 6·53-s + 5·55-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.753 − 1.30i)11-s − 0.727·17-s + 1.14·19-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + (−0.928 − 1.60i)29-s + (0.179 − 0.311i)31-s + 0.657·37-s + (−0.234 + 0.405i)41-s + (−0.228 − 0.396i)43-s + (0.291 + 0.505i)47-s + (0.5 − 0.866i)49-s + 0.824·53-s + 0.674·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147481398\)
\(L(\frac12)\) \(\approx\) \(1.147481398\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5 + 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.5 + 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.737554885136418086752983568606, −8.788706251432360872705215774460, −8.049922546873995106559747221301, −7.26834966634586408410464585410, −6.25184013947492097002411658442, −5.51186283950166454204954400442, −4.39603607525990205970093455395, −3.29675854143894482553301155332, −2.41506223502220968744004461809, −0.52786423627989410920559634578, 1.44122129871668389146920094671, 2.74230156776671143086075635522, 3.94209518255421705959660601965, 4.98099160613752284386215235684, 5.54365566509771538340622420165, 7.07016566737228500032490693960, 7.36291071449708716212659163074, 8.448724820006663224210405519239, 9.315041163704273039579873028536, 9.923563652643577875310518289806

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