Properties

Label 2-33e2-3.2-c2-0-41
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.80i·2-s − 3.87·4-s − 2.03i·5-s − 0.504·7-s + 0.356i·8-s + 5.70·10-s + 22.0·13-s − 1.41i·14-s − 16.4·16-s − 11.9i·17-s − 5.92·19-s + 7.86i·20-s − 42.7i·23-s + 20.8·25-s + 61.9i·26-s + ⋯
L(s)  = 1  + 1.40i·2-s − 0.968·4-s − 0.406i·5-s − 0.0720·7-s + 0.0445i·8-s + 0.570·10-s + 1.69·13-s − 0.101i·14-s − 1.03·16-s − 0.704i·17-s − 0.311·19-s + 0.393i·20-s − 1.85i·23-s + 0.834·25-s + 2.38i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.930330095\)
\(L(\frac12)\) \(\approx\) \(1.930330095\)
\(L(2)\) 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 \( 1 - 2.80iT - 4T^{2} \)
5 \( 1 + 2.03iT - 25T^{2} \)
7 \( 1 + 0.504T + 49T^{2} \)
13 \( 1 - 22.0T + 169T^{2} \)
17 \( 1 + 11.9iT - 289T^{2} \)
19 \( 1 + 5.92T + 361T^{2} \)
23 \( 1 + 42.7iT - 529T^{2} \)
29 \( 1 + 36.4iT - 841T^{2} \)
31 \( 1 + 3.49T + 961T^{2} \)
37 \( 1 + 32.8T + 1.36e3T^{2} \)
41 \( 1 + 27.2iT - 1.68e3T^{2} \)
43 \( 1 - 58.0T + 1.84e3T^{2} \)
47 \( 1 + 11.8iT - 2.20e3T^{2} \)
53 \( 1 - 61.7iT - 2.80e3T^{2} \)
59 \( 1 - 59.9iT - 3.48e3T^{2} \)
61 \( 1 - 49.5T + 3.72e3T^{2} \)
67 \( 1 - 108.T + 4.48e3T^{2} \)
71 \( 1 - 51.4iT - 5.04e3T^{2} \)
73 \( 1 - 123.T + 5.32e3T^{2} \)
79 \( 1 - 9.12T + 6.24e3T^{2} \)
83 \( 1 - 88.2iT - 6.88e3T^{2} \)
89 \( 1 + 4.52iT - 7.92e3T^{2} \)
97 \( 1 - 71.1T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.407533753888807838033275354215, −8.569334262125757888461615742699, −8.293710881287694166082779462831, −7.15021175914762608884891873403, −6.44587428026018575114057930954, −5.77922629242179117187116977317, −4.82638159116641483917915353404, −3.95700598650216172299074139179, −2.43853256572762019495544651512, −0.71287368430350526799616229463, 1.09465044390140256520129479410, 1.98477896599178884780928844724, 3.46544409900859677498883544901, 3.57871185876543514126968879247, 5.01359089791674693840793189009, 6.18521053390602217384830818240, 6.94990787589120012217214412360, 8.170236012466068510398913138979, 8.993729722474859411090233341290, 9.731135062729649868763304717493

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