Properties

Label 2-33e2-3.2-c2-0-25
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  + 1.35i·2-s + 2.17·4-s − 5.10i·5-s − 13.1·7-s + 8.34i·8-s + 6.89·10-s − 2.67·13-s − 17.7i·14-s − 2.58·16-s + 17.2i·17-s + 30.6·19-s − 11.0i·20-s − 15.5i·23-s − 1.05·25-s − 3.60i·26-s + ⋯
L(s)  = 1  + 0.675i·2-s + 0.543·4-s − 1.02i·5-s − 1.88·7-s + 1.04i·8-s + 0.689·10-s − 0.205·13-s − 1.27i·14-s − 0.161·16-s + 1.01i·17-s + 1.61·19-s − 0.554i·20-s − 0.675i·23-s − 0.0423·25-s − 0.138i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.787361311\)
\(L(\frac12)\) \(\approx\) \(1.787361311\)
\(L(2)\) 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 \( 1 - 1.35iT - 4T^{2} \)
5 \( 1 + 5.10iT - 25T^{2} \)
7 \( 1 + 13.1T + 49T^{2} \)
13 \( 1 + 2.67T + 169T^{2} \)
17 \( 1 - 17.2iT - 289T^{2} \)
19 \( 1 - 30.6T + 361T^{2} \)
23 \( 1 + 15.5iT - 529T^{2} \)
29 \( 1 + 10.3iT - 841T^{2} \)
31 \( 1 - 16.8T + 961T^{2} \)
37 \( 1 - 18.7T + 1.36e3T^{2} \)
41 \( 1 + 13.3iT - 1.68e3T^{2} \)
43 \( 1 - 10.5T + 1.84e3T^{2} \)
47 \( 1 - 82.2iT - 2.20e3T^{2} \)
53 \( 1 - 62.9iT - 2.80e3T^{2} \)
59 \( 1 + 65.9iT - 3.48e3T^{2} \)
61 \( 1 - 36.5T + 3.72e3T^{2} \)
67 \( 1 - 60.5T + 4.48e3T^{2} \)
71 \( 1 - 49.1iT - 5.04e3T^{2} \)
73 \( 1 - 6.37T + 5.32e3T^{2} \)
79 \( 1 - 115.T + 6.24e3T^{2} \)
83 \( 1 + 40.4iT - 6.88e3T^{2} \)
89 \( 1 + 71.1iT - 7.92e3T^{2} \)
97 \( 1 - 18.6T + 9.40e3T^{2} \)
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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

−9.647817971484157047152863712200, −8.971744462273135296379155301468, −8.049315136469241306478403937501, −7.22864289315838725218769592529, −6.32526904456281792403383718407, −5.84293138949568250851089533579, −4.79463147612271146939839516081, −3.52412379590895416101403928094, −2.54447959345324781670723812276, −0.914615974142711269722602387389, 0.70131519973013443589583244889, 2.45424737332794041562389712664, 3.14293137738252905415735558997, 3.63287861259431312084678904064, 5.38408072575604192183961661687, 6.44079486079713395606257171885, 6.94688190177563036675986592913, 7.53482776488354546801118093710, 9.175873429737100513856120591582, 9.879031373093700219471209583537

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