Properties

Label 2-33e2-3.2-c2-0-67
Degree $2$
Conductor $1089$
Sign $-0.577 - 0.816i$
Analytic cond. $29.6731$
Root an. cond. $5.44730$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.60i·2-s + 1.41·4-s − 6.21i·5-s − 11.1·7-s − 8.70i·8-s − 9.98·10-s + 17.7·13-s + 17.9i·14-s − 8.31·16-s − 8.53i·17-s − 16.4·19-s − 8.80i·20-s − 25.0i·23-s − 13.5·25-s − 28.5i·26-s + ⋯
L(s)  = 1  − 0.803i·2-s + 0.354·4-s − 1.24i·5-s − 1.59·7-s − 1.08i·8-s − 0.998·10-s + 1.36·13-s + 1.27i·14-s − 0.519·16-s − 0.502i·17-s − 0.866·19-s − 0.440i·20-s − 1.09i·23-s − 0.543·25-s − 1.09i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9929842646\)
\(L(\frac12)\) \(\approx\) \(0.9929842646\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + 1.60iT - 4T^{2} \)
5 \( 1 + 6.21iT - 25T^{2} \)
7 \( 1 + 11.1T + 49T^{2} \)
13 \( 1 - 17.7T + 169T^{2} \)
17 \( 1 + 8.53iT - 289T^{2} \)
19 \( 1 + 16.4T + 361T^{2} \)
23 \( 1 + 25.0iT - 529T^{2} \)
29 \( 1 + 9.46iT - 841T^{2} \)
31 \( 1 + 46.1T + 961T^{2} \)
37 \( 1 + 37.4T + 1.36e3T^{2} \)
41 \( 1 - 51.8iT - 1.68e3T^{2} \)
43 \( 1 - 55.6T + 1.84e3T^{2} \)
47 \( 1 - 37.8iT - 2.20e3T^{2} \)
53 \( 1 - 58.5iT - 2.80e3T^{2} \)
59 \( 1 + 44.6iT - 3.48e3T^{2} \)
61 \( 1 - 10.1T + 3.72e3T^{2} \)
67 \( 1 + 58.3T + 4.48e3T^{2} \)
71 \( 1 - 75.1iT - 5.04e3T^{2} \)
73 \( 1 + 112.T + 5.32e3T^{2} \)
79 \( 1 + 52.2T + 6.24e3T^{2} \)
83 \( 1 + 18.0iT - 6.88e3T^{2} \)
89 \( 1 + 136. iT - 7.92e3T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−9.095474389182379903667363637585, −8.775059414464044771092325083843, −7.43213318872196013253899630659, −6.41262173273079685412555043457, −5.90730344318992853398573615057, −4.45627416852249953187167695337, −3.63157585435489495355779529089, −2.68996916129519062626356218950, −1.36342041027898595886336821621, −0.29113404827464644187027136212, 1.99841489883646292178510673962, 3.20552549403357173148217824936, 3.75702765973370736733531876199, 5.69111604798454262486085845730, 6.09236963332343341703504499045, 6.92735711899597938797907710216, 7.26855753483584990430123014627, 8.515081230865917396765457029079, 9.237895495899780944406041154302, 10.52949249252658712157281870883

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