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Properties

Label 2-33e2-11.10-c2-0-45
Degree $2$
Conductor $1089$
Sign $0.904 - 0.426i$
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  + 3.11i·2-s − 5.68·4-s − 5.66·5-s + 9.60i·7-s − 5.24i·8-s − 17.6i·10-s − 21.3i·13-s − 29.8·14-s − 6.41·16-s + 16.6i·17-s − 23.1i·19-s + 32.2·20-s − 34.0·23-s + 7.14·25-s + 66.4·26-s + ⋯
L(s)  = 1  + 1.55i·2-s − 1.42·4-s − 1.13·5-s + 1.37i·7-s − 0.655i·8-s − 1.76i·10-s − 1.64i·13-s − 2.13·14-s − 0.400·16-s + 0.976i·17-s − 1.21i·19-s + 1.61·20-s − 1.48·23-s + 0.285·25-s + 2.55·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.904 - 0.426i$
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: \((2,\ 1089,\ (\ :1),\ 0.904 - 0.426i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5415782810\)
\(L(\frac12)\) \(\approx\) \(0.5415782810\)
\(L(2)\) 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 \( 1 - 3.11iT - 4T^{2} \)
5 \( 1 + 5.66T + 25T^{2} \)
7 \( 1 - 9.60iT - 49T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 - 16.6iT - 289T^{2} \)
19 \( 1 + 23.1iT - 361T^{2} \)
23 \( 1 + 34.0T + 529T^{2} \)
29 \( 1 - 17.2iT - 841T^{2} \)
31 \( 1 - 22.9T + 961T^{2} \)
37 \( 1 - 3.52T + 1.36e3T^{2} \)
41 \( 1 + 4.44iT - 1.68e3T^{2} \)
43 \( 1 + 26.1iT - 1.84e3T^{2} \)
47 \( 1 - 53.5T + 2.20e3T^{2} \)
53 \( 1 + 2.80T + 2.80e3T^{2} \)
59 \( 1 + 52.5T + 3.48e3T^{2} \)
61 \( 1 + 79.5iT - 3.72e3T^{2} \)
67 \( 1 - 61.8T + 4.48e3T^{2} \)
71 \( 1 + 11.8T + 5.04e3T^{2} \)
73 \( 1 - 76.3iT - 5.32e3T^{2} \)
79 \( 1 - 85.4iT - 6.24e3T^{2} \)
83 \( 1 + 44.6iT - 6.88e3T^{2} \)
89 \( 1 - 145.T + 7.92e3T^{2} \)
97 \( 1 - 96.9T + 9.40e3T^{2} \)
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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

−9.296045156358776182837091215681, −8.396405938424688726814963539263, −8.145923272998983582468761608688, −7.34932344949917852714854316504, −6.29399262937161469469917980952, −5.64832654236829330381819621797, −4.87867576935045017005871195183, −3.75592435114507034541646351931, −2.50589813970152503663601978446, −0.21164492187786101036792605464, 0.932114789020513137419886919599, 2.10685987617615277631066301539, 3.47450173688858867176747282683, 4.14208470064562614008856067171, 4.51516210078302467158487400577, 6.34320312499450509173878607235, 7.34636570330988071605908482196, 7.978125134629876577068230815929, 9.124656128792631880014835900956, 9.906217629146380363368551091801

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