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Properties

Label 2-33e2-11.10-c2-0-47
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  + 0.313i·2-s + 3.90·4-s + 7.41·5-s + 10.0i·7-s + 2.47i·8-s + 2.32i·10-s + 3.40i·13-s − 3.13·14-s + 14.8·16-s − 15.5i·17-s + 30.6i·19-s + 28.9·20-s − 7.67·23-s + 29.9·25-s − 1.06·26-s + ⋯
L(s)  = 1  + 0.156i·2-s + 0.975·4-s + 1.48·5-s + 1.43i·7-s + 0.309i·8-s + 0.232i·10-s + 0.261i·13-s − 0.224·14-s + 0.927·16-s − 0.915i·17-s + 1.61i·19-s + 1.44·20-s − 0.333·23-s + 1.19·25-s − 0.0410·26-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

\(L(\frac{3}{2})\) \(\approx\) \(3.295552841\)
\(L(\frac12)\) \(\approx\) \(3.295552841\)
\(L(2)\) 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 \( 1 - 0.313iT - 4T^{2} \)
5 \( 1 - 7.41T + 25T^{2} \)
7 \( 1 - 10.0iT - 49T^{2} \)
13 \( 1 - 3.40iT - 169T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 - 30.6iT - 361T^{2} \)
23 \( 1 + 7.67T + 529T^{2} \)
29 \( 1 - 3.37iT - 841T^{2} \)
31 \( 1 + 4.04T + 961T^{2} \)
37 \( 1 + 2.37T + 1.36e3T^{2} \)
41 \( 1 + 7.03iT - 1.68e3T^{2} \)
43 \( 1 + 3.99iT - 1.84e3T^{2} \)
47 \( 1 + 49.4T + 2.20e3T^{2} \)
53 \( 1 - 59.7T + 2.80e3T^{2} \)
59 \( 1 - 10.9T + 3.48e3T^{2} \)
61 \( 1 + 74.3iT - 3.72e3T^{2} \)
67 \( 1 + 3.22T + 4.48e3T^{2} \)
71 \( 1 + 116.T + 5.04e3T^{2} \)
73 \( 1 + 18.7iT - 5.32e3T^{2} \)
79 \( 1 + 3.51iT - 6.24e3T^{2} \)
83 \( 1 + 147. iT - 6.88e3T^{2} \)
89 \( 1 - 65.8T + 7.92e3T^{2} \)
97 \( 1 - 64.8T + 9.40e3T^{2} \)
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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

−9.841691752642861350832413110212, −9.091494443434523825372123338728, −8.246051732476550176184936896289, −7.19152020601194351961204434329, −6.14163085871905939486646207514, −5.86136121003049103733012150978, −5.04342438574268404809343870818, −3.23388141710186311369968771414, −2.24907560773612888953570509197, −1.69240746883897660600328481376, 0.975170209889002783990666921105, 1.97112130557277363050711723959, 2.97927555171129074273279919439, 4.17403160081076833615505759707, 5.37736018358825789233057744521, 6.28227386814735333293814809406, 6.88303673720353327979461845144, 7.65232682686425313413031124266, 8.802513149239386637314029324630, 9.861882988443285507578416274801

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