↑

Properties

Label 2-33e2-3.2-c2-0-61
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  + 3.75i·2-s − 10.0·4-s − 7.83i·5-s + 8.18·7-s − 22.8i·8-s + 29.4·10-s + 2.87·13-s + 30.6i·14-s + 45.3·16-s − 10.9i·17-s − 26.5·19-s + 79.0i·20-s + 17.9i·23-s − 36.4·25-s + 10.7i·26-s + ⋯
L(s)  = 1  + 1.87i·2-s − 2.52·4-s − 1.56i·5-s + 1.16·7-s − 2.85i·8-s + 2.94·10-s + 0.221·13-s + 2.19i·14-s + 2.83·16-s − 0.643i·17-s − 1.39·19-s + 3.95i·20-s + 0.781i·23-s − 1.45·25-s + 0.414i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.6414625949\)
\(L(\frac12)\) \(\approx\) \(0.6414625949\)
\(L(2)\) 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 \( 1 - 3.75iT - 4T^{2} \)
5 \( 1 + 7.83iT - 25T^{2} \)
7 \( 1 - 8.18T + 49T^{2} \)
13 \( 1 - 2.87T + 169T^{2} \)
17 \( 1 + 10.9iT - 289T^{2} \)
19 \( 1 + 26.5T + 361T^{2} \)
23 \( 1 - 17.9iT - 529T^{2} \)
29 \( 1 + 37.5iT - 841T^{2} \)
31 \( 1 + 31.0T + 961T^{2} \)
37 \( 1 + 45.4T + 1.36e3T^{2} \)
41 \( 1 - 57.8iT - 1.68e3T^{2} \)
43 \( 1 - 9.29T + 1.84e3T^{2} \)
47 \( 1 + 24.4iT - 2.20e3T^{2} \)
53 \( 1 + 32.6iT - 2.80e3T^{2} \)
59 \( 1 + 8.48iT - 3.48e3T^{2} \)
61 \( 1 + 91.7T + 3.72e3T^{2} \)
67 \( 1 + 40.8T + 4.48e3T^{2} \)
71 \( 1 - 123. iT - 5.04e3T^{2} \)
73 \( 1 - 85.7T + 5.32e3T^{2} \)
79 \( 1 + 65.2T + 6.24e3T^{2} \)
83 \( 1 + 39.7iT - 6.88e3T^{2} \)
89 \( 1 + 27.6iT - 7.92e3T^{2} \)
97 \( 1 - 36.2T + 9.40e3T^{2} \)
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

−9.073297309447745276448347384907, −8.440038106573029540516952520314, −8.020078764237145171376849919850, −7.16431719842269035363603956946, −6.04658372612420849667965056392, −5.29492811007661933152091367506, −4.69780904243611549264911691739, −4.04205335557734564007736833883, −1.56168166598395218433722889238, −0.19921488929492918187665252282, 1.62817435801681308075699775221, 2.32677650700877786892552541866, 3.36199643915309854267098979794, 4.12903283815530503492882794456, 5.12832985246856324825782406422, 6.35789897491072042241087152393, 7.51562403047894923875093541475, 8.505606406545281640518661009552, 9.134755953497749306019820801039, 10.48070331210425804764300542731

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