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Properties

Label 2-33e2-11.10-c2-0-84
Degree $2$
Conductor $1089$
Sign $0.372 - 0.927i$
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  − 2.21i·2-s − 0.906·4-s − 8.69·5-s − 6.67i·7-s − 6.85i·8-s + 19.2i·10-s − 11.1i·13-s − 14.7·14-s − 18.8·16-s − 7.82i·17-s − 8.44i·19-s + 7.88·20-s − 9.30·23-s + 50.6·25-s − 24.7·26-s + ⋯
L(s)  = 1  − 1.10i·2-s − 0.226·4-s − 1.73·5-s − 0.953i·7-s − 0.856i·8-s + 1.92i·10-s − 0.858i·13-s − 1.05·14-s − 1.17·16-s − 0.460i·17-s − 0.444i·19-s + 0.394·20-s − 0.404·23-s + 2.02·25-s − 0.950·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.372 - 0.927i$
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: \((2,\ 1089,\ (\ :1),\ 0.372 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3721683176\)
\(L(\frac12)\) \(\approx\) \(0.3721683176\)
\(L(2)\) 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 \( 1 + 2.21iT - 4T^{2} \)
5 \( 1 + 8.69T + 25T^{2} \)
7 \( 1 + 6.67iT - 49T^{2} \)
13 \( 1 + 11.1iT - 169T^{2} \)
17 \( 1 + 7.82iT - 289T^{2} \)
19 \( 1 + 8.44iT - 361T^{2} \)
23 \( 1 + 9.30T + 529T^{2} \)
29 \( 1 - 7.17iT - 841T^{2} \)
31 \( 1 + 27.4T + 961T^{2} \)
37 \( 1 + 52.9T + 1.36e3T^{2} \)
41 \( 1 + 47.8iT - 1.68e3T^{2} \)
43 \( 1 - 45.8iT - 1.84e3T^{2} \)
47 \( 1 - 16.0T + 2.20e3T^{2} \)
53 \( 1 - 54.9T + 2.80e3T^{2} \)
59 \( 1 - 71.2T + 3.48e3T^{2} \)
61 \( 1 - 1.30iT - 3.72e3T^{2} \)
67 \( 1 + 28.9T + 4.48e3T^{2} \)
71 \( 1 + 22.0T + 5.04e3T^{2} \)
73 \( 1 - 125. iT - 5.32e3T^{2} \)
79 \( 1 - 42.8iT - 6.24e3T^{2} \)
83 \( 1 - 133. iT - 6.88e3T^{2} \)
89 \( 1 - 9.48T + 7.92e3T^{2} \)
97 \( 1 - 156.T + 9.40e3T^{2} \)
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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

−9.105416885923460507162691322360, −8.157747655523577642341690531330, −7.29749077083399892484903055334, −6.90477240755607612828076465544, −5.22417725767605492646067851969, −4.03223591909523398244765480875, −3.68900050278936235712423944094, −2.66646658338507275309580453020, −1.01406075616210308193756737431, −0.13441836168448476663071239206, 2.04614038432663936773178679458, 3.44544398132813238477289388098, 4.38590736273279198902419264259, 5.36117466907122709289593135942, 6.26811579781848671314267517971, 7.15443771263763496962666214168, 7.72029686750817018612415559390, 8.595088866161590533483793379277, 8.917473006070644449954297669526, 10.40659970407582733888965723197

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