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Properties

Label 2-33e2-11.10-c2-0-11
Degree $2$
Conductor $1089$
Sign $-0.522 + 0.852i$
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  + 2.50i·2-s − 2.26·4-s + 1.29·5-s + 7.07i·7-s + 4.33i·8-s + 3.24i·10-s + 4.48i·13-s − 17.7·14-s − 19.9·16-s − 9.83i·17-s − 16.0i·19-s − 2.93·20-s − 39.6·23-s − 23.3·25-s − 11.2·26-s + ⋯
L(s)  = 1  + 1.25i·2-s − 0.566·4-s + 0.259·5-s + 1.01i·7-s + 0.542i·8-s + 0.324i·10-s + 0.344i·13-s − 1.26·14-s − 1.24·16-s − 0.578i·17-s − 0.842i·19-s − 0.146·20-s − 1.72·23-s − 0.932·25-s − 0.431·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.522 + 0.852i$
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: \((2,\ 1089,\ (\ :1),\ -0.522 + 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.085728677\)
\(L(\frac12)\) \(\approx\) \(1.085728677\)
\(L(2)\) 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 \( 1 - 2.50iT - 4T^{2} \)
5 \( 1 - 1.29T + 25T^{2} \)
7 \( 1 - 7.07iT - 49T^{2} \)
13 \( 1 - 4.48iT - 169T^{2} \)
17 \( 1 + 9.83iT - 289T^{2} \)
19 \( 1 + 16.0iT - 361T^{2} \)
23 \( 1 + 39.6T + 529T^{2} \)
29 \( 1 - 46.7iT - 841T^{2} \)
31 \( 1 - 34T + 961T^{2} \)
37 \( 1 - 11.3T + 1.36e3T^{2} \)
41 \( 1 - 65.0iT - 1.68e3T^{2} \)
43 \( 1 - 36.7iT - 1.84e3T^{2} \)
47 \( 1 + 78.5T + 2.20e3T^{2} \)
53 \( 1 - 29.3T + 2.80e3T^{2} \)
59 \( 1 + 66.0T + 3.48e3T^{2} \)
61 \( 1 + 97.2iT - 3.72e3T^{2} \)
67 \( 1 - 39.0T + 4.48e3T^{2} \)
71 \( 1 - 43.6T + 5.04e3T^{2} \)
73 \( 1 + 75.3iT - 5.32e3T^{2} \)
79 \( 1 + 61.0iT - 6.24e3T^{2} \)
83 \( 1 - 61.0iT - 6.88e3T^{2} \)
89 \( 1 + 103.T + 7.92e3T^{2} \)
97 \( 1 - 10.6T + 9.40e3T^{2} \)
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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

−9.840865921944250252336862684680, −9.243754887380848997436012618084, −8.335944338432256239350401648398, −7.79268294722070086245805119489, −6.65016927957876614147167667414, −6.19952295803730470562832567224, −5.28867607541889199426590563713, −4.56315996274293926536275694651, −2.94644394713143350090888127313, −1.90102183111061996790806992638, 0.30778130149894291176370926157, 1.55799612329449811640995115753, 2.48746364400982933562343421074, 3.88178003586882538756718581423, 4.09917873568460035175121825024, 5.70256474266205385690065786981, 6.52433998329324099709366327326, 7.62923266516081555746459359795, 8.329124515240819588649376428613, 9.662944331400626514337949856953

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