Properties

Label 2-33e2-3.2-c2-0-28
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  + 2.03i·2-s − 0.127·4-s − 0.796i·5-s + 1.12·7-s + 7.86i·8-s + 1.61·10-s + 9.25·13-s + 2.28i·14-s − 16.4·16-s + 11.7i·17-s + 17.2·19-s + 0.101i·20-s + 5.11i·23-s + 24.3·25-s + 18.7i·26-s + ⋯
L(s)  = 1  + 1.01i·2-s − 0.0317·4-s − 0.159i·5-s + 0.161·7-s + 0.983i·8-s + 0.161·10-s + 0.711·13-s + 0.163i·14-s − 1.03·16-s + 0.691i·17-s + 0.907·19-s + 0.00506i·20-s + 0.222i·23-s + 0.974·25-s + 0.723i·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

\(L(\frac{3}{2})\) \(\approx\) \(2.218807024\)
\(L(\frac12)\) \(\approx\) \(2.218807024\)
\(L(2)\) 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 \( 1 - 2.03iT - 4T^{2} \)
5 \( 1 + 0.796iT - 25T^{2} \)
7 \( 1 - 1.12T + 49T^{2} \)
13 \( 1 - 9.25T + 169T^{2} \)
17 \( 1 - 11.7iT - 289T^{2} \)
19 \( 1 - 17.2T + 361T^{2} \)
23 \( 1 - 5.11iT - 529T^{2} \)
29 \( 1 + 29.0iT - 841T^{2} \)
31 \( 1 + 52.7T + 961T^{2} \)
37 \( 1 - 37.3T + 1.36e3T^{2} \)
41 \( 1 - 75.0iT - 1.68e3T^{2} \)
43 \( 1 - 68.8T + 1.84e3T^{2} \)
47 \( 1 - 9.36iT - 2.20e3T^{2} \)
53 \( 1 + 34.0iT - 2.80e3T^{2} \)
59 \( 1 - 69.2iT - 3.48e3T^{2} \)
61 \( 1 + 24.9T + 3.72e3T^{2} \)
67 \( 1 + 16.8T + 4.48e3T^{2} \)
71 \( 1 - 77.0iT - 5.04e3T^{2} \)
73 \( 1 - 55.2T + 5.32e3T^{2} \)
79 \( 1 - 64.9T + 6.24e3T^{2} \)
83 \( 1 - 120. iT - 6.88e3T^{2} \)
89 \( 1 - 110. iT - 7.92e3T^{2} \)
97 \( 1 + 14.0T + 9.40e3T^{2} \)
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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

−9.743267452171918107243365327855, −8.934396497701073461979033432646, −8.071560499136874244042512489399, −7.54147887558552401734663583693, −6.54315366785966387002567814429, −5.87088183753429079739816207087, −5.07283945756648761354538654417, −3.95885043340886240044830897365, −2.66940524547278730847312609424, −1.30247469953930932776155304065, 0.73131308295227474162414381945, 1.86391409557136897611391869061, 2.99172221056120840119533005485, 3.71702711256395072775408195565, 4.89426609159045627654247269654, 5.95714051890403671309830757112, 6.99375236967548360220096926215, 7.61175330467820592838128846579, 8.964526317461799485126371958617, 9.398819544378404901114651922366

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