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Properties

Label 2-33e2-11.10-c2-0-66
Degree $2$
Conductor $1089$
Sign $0.372 + 0.927i$
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  + 2.88i·2-s − 4.32·4-s + 0.441·5-s − 10.5i·7-s − 0.927i·8-s + 1.27i·10-s − 6.44i·13-s + 30.5·14-s − 14.6·16-s + 18.8i·17-s + 27.5i·19-s − 1.90·20-s − 6.29·23-s − 24.8·25-s + 18.5·26-s + ⋯
L(s)  = 1  + 1.44i·2-s − 1.08·4-s + 0.0882·5-s − 1.51i·7-s − 0.115i·8-s + 0.127i·10-s − 0.495i·13-s + 2.18·14-s − 0.913·16-s + 1.11i·17-s + 1.44i·19-s − 0.0953·20-s − 0.273·23-s − 0.992·25-s + 0.715·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.372 + 0.927i$
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: \((2,\ 1089,\ (\ :1),\ 0.372 + 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3733355349\)
\(L(\frac12)\) \(\approx\) \(0.3733355349\)
\(L(2)\) 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 \( 1 - 2.88iT - 4T^{2} \)
5 \( 1 - 0.441T + 25T^{2} \)
7 \( 1 + 10.5iT - 49T^{2} \)
13 \( 1 + 6.44iT - 169T^{2} \)
17 \( 1 - 18.8iT - 289T^{2} \)
19 \( 1 - 27.5iT - 361T^{2} \)
23 \( 1 + 6.29T + 529T^{2} \)
29 \( 1 + 44.1iT - 841T^{2} \)
31 \( 1 + 27.1T + 961T^{2} \)
37 \( 1 + 0.853T + 1.36e3T^{2} \)
41 \( 1 + 13.2iT - 1.68e3T^{2} \)
43 \( 1 + 68.8iT - 1.84e3T^{2} \)
47 \( 1 - 16.1T + 2.20e3T^{2} \)
53 \( 1 + 38.6T + 2.80e3T^{2} \)
59 \( 1 + 76.9T + 3.48e3T^{2} \)
61 \( 1 + 28.6iT - 3.72e3T^{2} \)
67 \( 1 + 78.0T + 4.48e3T^{2} \)
71 \( 1 - 27.1T + 5.04e3T^{2} \)
73 \( 1 + 12.9iT - 5.32e3T^{2} \)
79 \( 1 + 94.2iT - 6.24e3T^{2} \)
83 \( 1 - 115. iT - 6.88e3T^{2} \)
89 \( 1 + 65.2T + 7.92e3T^{2} \)
97 \( 1 + 70.2T + 9.40e3T^{2} \)
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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

−9.401875290693735449375544942372, −8.153136443022850812637783946697, −7.85323047796675881372533033544, −7.08744192158541074582740301789, −6.13057044473559624227633745049, −5.61121750319160312923639916528, −4.30627684836930806368109400931, −3.70760576526943462298406635639, −1.79517740283026464992375387870, −0.10833044211374148545956943688, 1.53031276748863589147115362284, 2.52895843025918259108640171165, 3.13423948015562231613639081414, 4.48018340767423550571941143272, 5.28063095787739030680866628913, 6.37924888819283526048409385126, 7.36899464068372414224159601176, 8.702656574977490408558918761141, 9.269028684751024706602483268847, 9.700746930873088351937313080881

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