Properties

Label 2-33e2-33.32-c1-0-26
Degree $2$
Conductor $1089$
Sign $-0.972 - 0.231i$
Analytic cond. $8.69570$
Root an. cond. $2.94884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.43·2-s + 3.93·4-s − 3.79i·5-s − 0.367i·7-s − 4.71·8-s + 9.24i·10-s + 0.948i·13-s + 0.896i·14-s + 3.61·16-s − 3.43·17-s − 4.26i·19-s − 14.9i·20-s − 4.96i·23-s − 9.40·25-s − 2.31i·26-s + ⋯
L(s)  = 1  − 1.72·2-s + 1.96·4-s − 1.69i·5-s − 0.139i·7-s − 1.66·8-s + 2.92i·10-s + 0.263i·13-s + 0.239i·14-s + 0.904·16-s − 0.833·17-s − 0.977i·19-s − 3.34i·20-s − 1.03i·23-s − 1.88·25-s − 0.453i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.972 - 0.231i$
Analytic conductor: \(8.69570\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1088, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ -0.972 - 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2902085954\)
\(L(\frac12)\) \(\approx\) \(0.2902085954\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.43T + 2T^{2} \)
5 \( 1 + 3.79iT - 5T^{2} \)
7 \( 1 + 0.367iT - 7T^{2} \)
13 \( 1 - 0.948iT - 13T^{2} \)
17 \( 1 + 3.43T + 17T^{2} \)
19 \( 1 + 4.26iT - 19T^{2} \)
23 \( 1 + 4.96iT - 23T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 - 3.51T + 31T^{2} \)
37 \( 1 + 7.18T + 37T^{2} \)
41 \( 1 + 2.71T + 41T^{2} \)
43 \( 1 - 1.88iT - 43T^{2} \)
47 \( 1 + 0.0206iT - 47T^{2} \)
53 \( 1 - 5.54iT - 53T^{2} \)
59 \( 1 - 6.62iT - 59T^{2} \)
61 \( 1 + 9.75iT - 61T^{2} \)
67 \( 1 - 4.46T + 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 4.39iT - 73T^{2} \)
79 \( 1 + 10.9iT - 79T^{2} \)
83 \( 1 + 9.18T + 83T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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

−9.098465762272512833095743770795, −8.816613948352231354422520921037, −8.157391774747920734857841004557, −7.20852008654909486287121125540, −6.36938163211037379726028965745, −5.10226307252524744673554936567, −4.24956708819211652331820602001, −2.41893417288374219102995254190, −1.30492980399026857215308234693, −0.23600968729111307105706260439, 1.76134495542290762331392165882, 2.71084964315007959743580427594, 3.74460925830925112486475350559, 5.63560395915246864087765718958, 6.60631919215030036856128197388, 7.12191493927450181128236273420, 7.88087598729276182783857001913, 8.660251266215841459501006329769, 9.659825728375226606288910237872, 10.19116671087564725155100742111

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