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Properties

Label 2-33e2-9.4-c1-0-5
Degree $2$
Conductor $1089$
Sign $0.453 + 0.891i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.07 + 1.86i)2-s + (−0.635 + 1.61i)3-s + (−1.32 + 2.29i)4-s + (−1.81 + 3.13i)5-s + (−3.69 + 0.550i)6-s + (−1.13 − 1.96i)7-s − 1.39·8-s + (−2.19 − 2.04i)9-s − 7.81·10-s + (−2.85 − 3.58i)12-s + (0.619 − 1.07i)13-s + (2.44 − 4.23i)14-s + (−3.90 − 4.91i)15-s + (1.14 + 1.98i)16-s − 5.69·17-s + (1.45 − 6.30i)18-s + ⋯
L(s)  = 1  + (0.762 + 1.32i)2-s + (−0.366 + 0.930i)3-s + (−0.661 + 1.14i)4-s + (−0.810 + 1.40i)5-s + (−1.50 + 0.224i)6-s + (−0.429 − 0.743i)7-s − 0.492·8-s + (−0.730 − 0.682i)9-s − 2.47·10-s + (−0.823 − 1.03i)12-s + (0.171 − 0.297i)13-s + (0.654 − 1.13i)14-s + (−1.00 − 1.26i)15-s + (0.286 + 0.495i)16-s − 1.38·17-s + (0.343 − 1.48i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(8.69570\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ 0.453 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7884729834\)
\(L(\frac12)\) \(\approx\) \(0.7884729834\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.635 - 1.61i)T \)
11 \( 1 \)
good2 \( 1 + (-1.07 - 1.86i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.81 - 3.13i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.13 + 1.96i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-0.619 + 1.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.69T + 17T^{2} \)
19 \( 1 - 2.89T + 19T^{2} \)
23 \( 1 + (2.95 - 5.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.75 + 3.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.333T + 37T^{2} \)
41 \( 1 + (4.98 - 8.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.57 + 6.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.34 - 7.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.16T + 53T^{2} \)
59 \( 1 + (1.45 - 2.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.13 - 5.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.68 - 8.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 4.31T + 73T^{2} \)
79 \( 1 + (-0.708 - 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.37 + 2.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.77T + 89T^{2} \)
97 \( 1 + (-1.27 - 2.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49852060780443322625715665656, −9.916916487151501211942797353844, −8.667105874661911955537529186052, −7.63922677025566226068369068520, −7.06755827546943209866320969912, −6.34412230257438824021878454448, −5.60829839601018812321205623936, −4.37446289942802884649815432536, −3.87710263832594609031470472515, −3.02866272517221715681387709476, 0.28270308342861646940142264859, 1.56986184068512073044556605156, 2.54603019679810592487863051104, 3.79383824727963333923575082374, 4.74984398654325028100104808857, 5.36964601929225158422067097940, 6.47465930081412613088703279488, 7.57622576363808160402325423204, 8.637930916555716131104553021466, 9.032304459367122000653844347907

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