Properties

Label 2-33e2-3.2-c2-0-50
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  − 0.816i·2-s + 3.33·4-s + 1.04i·5-s + 6.83·7-s − 5.98i·8-s + 0.855·10-s − 7.70·13-s − 5.57i·14-s + 8.44·16-s − 9.39i·17-s + 9.42·19-s + 3.48i·20-s − 4.82i·23-s + 23.9·25-s + 6.29i·26-s + ⋯
L(s)  = 1  − 0.408i·2-s + 0.833·4-s + 0.209i·5-s + 0.975·7-s − 0.748i·8-s + 0.0855·10-s − 0.592·13-s − 0.398i·14-s + 0.527·16-s − 0.552i·17-s + 0.495·19-s + 0.174i·20-s − 0.209i·23-s + 0.956·25-s + 0.242i·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

\(L(\frac{3}{2})\) \(\approx\) \(2.781760202\)
\(L(\frac12)\) \(\approx\) \(2.781760202\)
\(L(2)\) 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 \( 1 + 0.816iT - 4T^{2} \)
5 \( 1 - 1.04iT - 25T^{2} \)
7 \( 1 - 6.83T + 49T^{2} \)
13 \( 1 + 7.70T + 169T^{2} \)
17 \( 1 + 9.39iT - 289T^{2} \)
19 \( 1 - 9.42T + 361T^{2} \)
23 \( 1 + 4.82iT - 529T^{2} \)
29 \( 1 + 46.7iT - 841T^{2} \)
31 \( 1 - 18.9T + 961T^{2} \)
37 \( 1 + 47.1T + 1.36e3T^{2} \)
41 \( 1 + 7.60iT - 1.68e3T^{2} \)
43 \( 1 - 50.6T + 1.84e3T^{2} \)
47 \( 1 - 66.1iT - 2.20e3T^{2} \)
53 \( 1 + 38.4iT - 2.80e3T^{2} \)
59 \( 1 - 117. iT - 3.48e3T^{2} \)
61 \( 1 - 81.4T + 3.72e3T^{2} \)
67 \( 1 - 70.1T + 4.48e3T^{2} \)
71 \( 1 - 43.2iT - 5.04e3T^{2} \)
73 \( 1 - 45.9T + 5.32e3T^{2} \)
79 \( 1 + 85.7T + 6.24e3T^{2} \)
83 \( 1 + 159. iT - 6.88e3T^{2} \)
89 \( 1 + 154. iT - 7.92e3T^{2} \)
97 \( 1 - 153.T + 9.40e3T^{2} \)
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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

−9.798413601117052392447132939721, −8.725078140169516825128632860788, −7.72282969033899613302264245450, −7.18710592593775945706310871517, −6.22203617046905608445926827964, −5.19835982354244041745318852133, −4.23416592832535609252896461775, −2.95482612528632181258338548807, −2.16733761285764479809092957271, −0.931181589979163827958812987567, 1.28137620415238939486028493012, 2.30464763078227721763787827287, 3.51445522522545704463781561112, 4.95213532631769754582753429600, 5.38844020765326056055185938353, 6.62872370870909019705756076595, 7.23246156705445401885136927335, 8.123570170547660873037899131698, 8.689382137456482671667620172775, 9.850937394611234421157020938680

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