Properties

Label 2-33e2-3.2-c2-0-30
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  + 3.25i·2-s − 6.59·4-s − 1.59i·5-s − 10.8·7-s − 8.43i·8-s + 5.18·10-s − 9.23·13-s − 35.4i·14-s + 1.07·16-s − 4.81i·17-s − 25.0·19-s + 10.4i·20-s + 36.6i·23-s + 22.4·25-s − 30.0i·26-s + ⋯
L(s)  = 1  + 1.62i·2-s − 1.64·4-s − 0.318i·5-s − 1.55·7-s − 1.05i·8-s + 0.518·10-s − 0.710·13-s − 2.52i·14-s + 0.0670·16-s − 0.283i·17-s − 1.32·19-s + 0.524i·20-s + 1.59i·23-s + 0.898·25-s − 1.15i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.8443955978\)
\(L(\frac12)\) \(\approx\) \(0.8443955978\)
\(L(2)\) 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 \( 1 - 3.25iT - 4T^{2} \)
5 \( 1 + 1.59iT - 25T^{2} \)
7 \( 1 + 10.8T + 49T^{2} \)
13 \( 1 + 9.23T + 169T^{2} \)
17 \( 1 + 4.81iT - 289T^{2} \)
19 \( 1 + 25.0T + 361T^{2} \)
23 \( 1 - 36.6iT - 529T^{2} \)
29 \( 1 + 24.4iT - 841T^{2} \)
31 \( 1 - 58.1T + 961T^{2} \)
37 \( 1 - 12.7T + 1.36e3T^{2} \)
41 \( 1 - 12.4iT - 1.68e3T^{2} \)
43 \( 1 - 19.6T + 1.84e3T^{2} \)
47 \( 1 + 64.9iT - 2.20e3T^{2} \)
53 \( 1 + 49.5iT - 2.80e3T^{2} \)
59 \( 1 - 24.0iT - 3.48e3T^{2} \)
61 \( 1 - 44.2T + 3.72e3T^{2} \)
67 \( 1 - 33.0T + 4.48e3T^{2} \)
71 \( 1 - 74.9iT - 5.04e3T^{2} \)
73 \( 1 - 41.3T + 5.32e3T^{2} \)
79 \( 1 - 63.9T + 6.24e3T^{2} \)
83 \( 1 + 42.1iT - 6.88e3T^{2} \)
89 \( 1 + 111. iT - 7.92e3T^{2} \)
97 \( 1 + 108.T + 9.40e3T^{2} \)
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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

−9.619017310511091967231556582665, −8.784903476085112321676904344106, −8.043642085782911385127994479551, −7.06075753118278227597360845741, −6.55294370357242928054115005812, −5.79413857802294381438543149905, −4.88508738551296979801223220751, −3.91462862481012430214863066455, −2.60353165038020520591164055045, −0.37845016224431688730461973825, 0.76545220777997325938297694865, 2.45553720994248026404350678925, 2.88215557222561923918819593118, 3.96802479603320235018505372155, 4.77476243558394662224693316771, 6.35096796923009763499933022601, 6.76865332468764086106218773958, 8.262420288659151999820215660898, 9.141386446265714646494505356279, 9.768697238329096566293515200970

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