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Properties

Label 2-33e2-99.52-c0-0-1
Degree $2$
Conductor $1089$
Sign $0.907 + 0.419i$
Analytic cond. $0.543481$
Root an. cond. $0.737212$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 − 0.147i)2-s + (−0.669 + 0.743i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)5-s + (−0.831 + 1.14i)6-s + (1.05 − 0.946i)7-s + (−0.104 − 0.994i)9-s − 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.33 − 1.48i)14-s + (0.669 + 0.743i)15-s + (−0.913 + 0.406i)16-s + (−0.294 − 1.38i)18-s + (−0.104 − 0.994i)20-s + 1.41i·21-s + ⋯
L(s)  = 1  + (1.40 − 0.147i)2-s + (−0.669 + 0.743i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)5-s + (−0.831 + 1.14i)6-s + (1.05 − 0.946i)7-s + (−0.104 − 0.994i)9-s − 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.33 − 1.48i)14-s + (0.669 + 0.743i)15-s + (−0.913 + 0.406i)16-s + (−0.294 − 1.38i)18-s + (−0.104 − 0.994i)20-s + 1.41i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.907 + 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.807360271\)
\(L(\frac12)\) \(\approx\) \(1.807360271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.669 - 0.743i)T \)
11 \( 1 \)
good2 \( 1 + (-1.40 + 0.147i)T + (0.978 - 0.207i)T^{2} \)
5 \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \)
7 \( 1 + (-1.05 + 0.946i)T + (0.104 - 0.994i)T^{2} \)
13 \( 1 + (-0.669 - 0.743i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.104 + 0.994i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
61 \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \)
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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

−10.32312631837784447430837677118, −9.222785081582814608186815812546, −8.485753091271703587158580715783, −7.27984244585279885121056249047, −6.21305649262040078837822011486, −5.29833215594350536790552376859, −4.73723898138493070210049796096, −4.25729626562213289455824036865, −3.23443371358814695295884240442, −1.39226227724889897118245719087, 2.04083923661676067610064735082, 2.81010196425896005412242348722, 4.19409936901548985177109479778, 5.16682573861426121142756408525, 5.79615797098757171750138937219, 6.45450964121942273182410332891, 7.31729781068745052315539492048, 8.136311313039771489039289459297, 9.284474300867174481892726273564, 10.59475991179198502760712034420

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