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Properties

Label 2-33e2-11.10-c2-0-50
Degree $2$
Conductor $1089$
Sign $0.522 - 0.852i$
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  + 0.517i·2-s + 3.73·4-s + 7.19·5-s + 8.86i·7-s + 4.00i·8-s + 3.72i·10-s − 16.0i·13-s − 4.58·14-s + 12.8·16-s + 27.0i·17-s − 13.3i·19-s + 26.8·20-s + 28.0·23-s + 26.7·25-s + 8.32·26-s + ⋯
L(s)  = 1  + 0.258i·2-s + 0.933·4-s + 1.43·5-s + 1.26i·7-s + 0.500i·8-s + 0.372i·10-s − 1.23i·13-s − 0.327·14-s + 0.803·16-s + 1.58i·17-s − 0.704i·19-s + 1.34·20-s + 1.21·23-s + 1.07·25-s + 0.320·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ 0.522 - 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.361785918\)
\(L(\frac12)\) \(\approx\) \(3.361785918\)
\(L(2)\) 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 \( 1 - 0.517iT - 4T^{2} \)
5 \( 1 - 7.19T + 25T^{2} \)
7 \( 1 - 8.86iT - 49T^{2} \)
13 \( 1 + 16.0iT - 169T^{2} \)
17 \( 1 - 27.0iT - 289T^{2} \)
19 \( 1 + 13.3iT - 361T^{2} \)
23 \( 1 - 28.0T + 529T^{2} \)
29 \( 1 - 3.79iT - 841T^{2} \)
31 \( 1 + 37.3T + 961T^{2} \)
37 \( 1 - 26.1T + 1.36e3T^{2} \)
41 \( 1 - 35.3iT - 1.68e3T^{2} \)
43 \( 1 + 22.5iT - 1.84e3T^{2} \)
47 \( 1 - 54.8T + 2.20e3T^{2} \)
53 \( 1 + 55.7T + 2.80e3T^{2} \)
59 \( 1 + 38.0T + 3.48e3T^{2} \)
61 \( 1 + 12.0iT - 3.72e3T^{2} \)
67 \( 1 + 81.9T + 4.48e3T^{2} \)
71 \( 1 - 52.6T + 5.04e3T^{2} \)
73 \( 1 + 129. iT - 5.32e3T^{2} \)
79 \( 1 - 59.7iT - 6.24e3T^{2} \)
83 \( 1 - 85.4iT - 6.88e3T^{2} \)
89 \( 1 + 123.T + 7.92e3T^{2} \)
97 \( 1 + 22.2T + 9.40e3T^{2} \)
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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

−9.775366737158099920859744435663, −8.975367987860827576386639470185, −8.231871252038692624885772620132, −7.18692370379769681770895207068, −6.09745908039917913191729971083, −5.84771683620337254757553966631, −5.06308056069894935396197078712, −3.15387096188920701337335765442, −2.39617545515612092647288261009, −1.50080423041066850889953268989, 1.06488172946944826791443733471, 1.97909617484023278833823403250, 2.99339157778196095541972346361, 4.20652051733565848351462487028, 5.34394734523080177040959428850, 6.28592505361340942611773858626, 7.04637115499782488251167127075, 7.50392042900972299206702837608, 9.107184382803796872577267410059, 9.605062626668872258737862136153

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