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Properties

Label 2-33e2-3.2-c2-0-43
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  + 3.62i·2-s − 9.10·4-s − 0.165i·5-s + 10.2·7-s − 18.4i·8-s + 0.598·10-s − 20.3·13-s + 37.2i·14-s + 30.5·16-s − 23.0i·17-s − 4.13·19-s + 1.50i·20-s − 13.5i·23-s + 24.9·25-s − 73.6i·26-s + ⋯
L(s)  = 1  + 1.81i·2-s − 2.27·4-s − 0.0330i·5-s + 1.47·7-s − 2.31i·8-s + 0.0598·10-s − 1.56·13-s + 2.66i·14-s + 1.90·16-s − 1.35i·17-s − 0.217·19-s + 0.0752i·20-s − 0.587i·23-s + 0.998·25-s − 2.83i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.475886032\)
\(L(\frac12)\) \(\approx\) \(1.475886032\)
\(L(2)\) 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 \( 1 - 3.62iT - 4T^{2} \)
5 \( 1 + 0.165iT - 25T^{2} \)
7 \( 1 - 10.2T + 49T^{2} \)
13 \( 1 + 20.3T + 169T^{2} \)
17 \( 1 + 23.0iT - 289T^{2} \)
19 \( 1 + 4.13T + 361T^{2} \)
23 \( 1 + 13.5iT - 529T^{2} \)
29 \( 1 + 4.01iT - 841T^{2} \)
31 \( 1 + 20.5T + 961T^{2} \)
37 \( 1 - 43.5T + 1.36e3T^{2} \)
41 \( 1 + 52.0iT - 1.68e3T^{2} \)
43 \( 1 - 62.5T + 1.84e3T^{2} \)
47 \( 1 + 38.7iT - 2.20e3T^{2} \)
53 \( 1 + 35.3iT - 2.80e3T^{2} \)
59 \( 1 + 19.6iT - 3.48e3T^{2} \)
61 \( 1 + 6.42T + 3.72e3T^{2} \)
67 \( 1 + 3.98T + 4.48e3T^{2} \)
71 \( 1 + 54.7iT - 5.04e3T^{2} \)
73 \( 1 + 22.8T + 5.32e3T^{2} \)
79 \( 1 + 57.1T + 6.24e3T^{2} \)
83 \( 1 - 56.8iT - 6.88e3T^{2} \)
89 \( 1 + 28.1iT - 7.92e3T^{2} \)
97 \( 1 - 82.9T + 9.40e3T^{2} \)
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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

−9.330102219886314608437438950158, −8.790303544675248449759046716211, −7.77866967214256013327701811590, −7.44199690991686691783020824274, −6.63439056726246779507682611436, −5.38554619422835517225585348360, −4.96521059152339138249359316347, −4.25841563520384497860632638986, −2.40823335419177358187042877728, −0.51845044468984750057909865632, 1.15089209807176981443150416925, 2.03421774846166311927508803780, 2.94080523233635655464948442551, 4.27978692301329421232085665001, 4.70185632201213811483330674339, 5.73918110613451852368902850628, 7.39707425917796620055611096628, 8.167701668198330423376134879956, 8.996197851180795057806885609632, 9.785194555966892078169170723339

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