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Properties

Label 2-33e2-3.2-c2-0-12
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  − 1.38i·2-s + 2.08·4-s − 0.765i·5-s − 8.89·7-s − 8.42i·8-s − 1.06·10-s − 10.8·13-s + 12.3i·14-s − 3.33·16-s + 20.8i·17-s − 9.79·19-s − 1.59i·20-s + 25.0i·23-s + 24.4·25-s + 15.0i·26-s + ⋯
L(s)  = 1  − 0.692i·2-s + 0.520·4-s − 0.153i·5-s − 1.27·7-s − 1.05i·8-s − 0.106·10-s − 0.834·13-s + 0.879i·14-s − 0.208·16-s + 1.22i·17-s − 0.515·19-s − 0.0797i·20-s + 1.08i·23-s + 0.976·25-s + 0.577i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.012722975\)
\(L(\frac12)\) \(\approx\) \(1.012722975\)
\(L(2)\) 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 \( 1 + 1.38iT - 4T^{2} \)
5 \( 1 + 0.765iT - 25T^{2} \)
7 \( 1 + 8.89T + 49T^{2} \)
13 \( 1 + 10.8T + 169T^{2} \)
17 \( 1 - 20.8iT - 289T^{2} \)
19 \( 1 + 9.79T + 361T^{2} \)
23 \( 1 - 25.0iT - 529T^{2} \)
29 \( 1 - 13.8iT - 841T^{2} \)
31 \( 1 + 18.9T + 961T^{2} \)
37 \( 1 - 15.4T + 1.36e3T^{2} \)
41 \( 1 - 62.8iT - 1.68e3T^{2} \)
43 \( 1 - 74.8T + 1.84e3T^{2} \)
47 \( 1 - 67.4iT - 2.20e3T^{2} \)
53 \( 1 + 96.3iT - 2.80e3T^{2} \)
59 \( 1 - 8.48iT - 3.48e3T^{2} \)
61 \( 1 + 93.4T + 3.72e3T^{2} \)
67 \( 1 - 80.8T + 4.48e3T^{2} \)
71 \( 1 - 91.2iT - 5.04e3T^{2} \)
73 \( 1 + 15.9T + 5.32e3T^{2} \)
79 \( 1 + 83.6T + 6.24e3T^{2} \)
83 \( 1 - 98.8iT - 6.88e3T^{2} \)
89 \( 1 - 113. iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−9.745421157592985614642209713184, −9.415559268368476759308536358575, −8.167017357581441095846566070561, −7.15957416027472743145657012255, −6.49018243625189865854421019632, −5.67035872288575291180983235619, −4.27136270543244418255238146505, −3.34001346787662941641512878864, −2.53426735659201865352293565872, −1.27569522771157893800500179692, 0.29933596790342200976634197502, 2.35912035664658074078628633865, 3.01106567962722941272855378345, 4.43373998997165492856796507849, 5.51302423031270702201806721094, 6.32864413787756987926396842986, 7.04364104891584985140436365811, 7.53134920058495544886274185473, 8.755395355253786206909011775328, 9.414766846179250093898059634057

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