Properties

Label 2-33e2-9.7-c1-0-98
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: $\chi_{1089} (727, \cdot )$
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

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

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