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Properties

Label 2-1080-72.11-c1-0-37
Degree $2$
Conductor $1080$
Sign $-0.827 + 0.561i$
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  + (−0.962 − 1.03i)2-s + (−0.146 + 1.99i)4-s + (0.5 + 0.866i)5-s + (1.68 + 0.970i)7-s + (2.20 − 1.76i)8-s + (0.415 − 1.35i)10-s + (−4.90 − 2.83i)11-s + (−4.57 + 2.63i)13-s + (−0.612 − 2.67i)14-s + (−3.95 − 0.584i)16-s − 1.68i·17-s − 1.99·19-s + (−1.80 + 0.870i)20-s + (1.78 + 7.80i)22-s + (−3.68 − 6.38i)23-s + ⋯
L(s)  = 1  + (−0.680 − 0.732i)2-s + (−0.0733 + 0.997i)4-s + (0.223 + 0.387i)5-s + (0.635 + 0.366i)7-s + (0.780 − 0.625i)8-s + (0.131 − 0.427i)10-s + (−1.47 − 0.853i)11-s + (−1.26 + 0.731i)13-s + (−0.163 − 0.715i)14-s + (−0.989 − 0.146i)16-s − 0.409i·17-s − 0.456·19-s + (−0.402 + 0.194i)20-s + (0.380 + 1.66i)22-s + (−0.768 − 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5422556835\)
\(L(\frac12)\) \(\approx\) \(0.5422556835\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.962 + 1.03i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-1.68 - 0.970i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.90 + 2.83i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.57 - 2.63i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.68iT - 17T^{2} \)
19 \( 1 + 1.99T + 19T^{2} \)
23 \( 1 + (3.68 + 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.72 + 2.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.12 + 5.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.70iT - 37T^{2} \)
41 \( 1 + (-0.985 + 0.569i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.18 + 3.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.78 + 3.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 + (4.64 - 2.68i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.02 - 4.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.52 + 4.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.47T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + (-2.53 - 1.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.94 - 2.85i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (-1.57 + 2.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.723506676180241610407603442250, −8.708952579601928632574514088699, −8.058471678827323224663059230072, −7.35991811841430013578928514845, −6.24119144736750479727555198014, −5.05863565631225931181906454117, −4.16146441305969506736110809551, −2.54794373973190233132149462615, −2.35477173408031779444014883643, −0.29504153584400502402696627245, 1.44870735211164863974031774026, 2.66092323817912143332186057709, 4.69178010866225481461377632461, 4.97955265425377306858150607877, 6.00793609988200946971305533850, 7.13585798885134534801878515307, 7.902062152581128804191846293281, 8.203343613404105829661810576208, 9.457436432602412010788317775266, 10.20395423511980325984392253136

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