Properties

Label 2-33e2-99.40-c0-0-0
Degree $2$
Conductor $1089$
Sign $-0.197 - 0.980i$
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  + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.104 + 0.994i)9-s + (−0.5 − 0.866i)12-s + (−0.669 + 0.743i)15-s + (0.913 + 0.406i)16-s + (0.104 − 0.994i)20-s + (−1 + 1.73i)23-s + (−0.809 + 0.587i)27-s + (−0.913 + 0.406i)31-s + (0.309 − 0.951i)36-s + (−0.309 − 0.951i)37-s − 45-s + (0.978 − 0.207i)47-s + (0.309 + 0.951i)48-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.104 + 0.994i)9-s + (−0.5 − 0.866i)12-s + (−0.669 + 0.743i)15-s + (0.913 + 0.406i)16-s + (0.104 − 0.994i)20-s + (−1 + 1.73i)23-s + (−0.809 + 0.587i)27-s + (−0.913 + 0.406i)31-s + (0.309 − 0.951i)36-s + (−0.309 − 0.951i)37-s − 45-s + (0.978 − 0.207i)47-s + (0.309 + 0.951i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.197 - 0.980i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.197 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9832405106\)
\(L(\frac12)\) \(\approx\) \(0.9832405106\)
\(L(1)\) 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 \( 1 + (0.978 + 0.207i)T^{2} \)
5 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
7 \( 1 + (0.104 + 0.994i)T^{2} \)
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 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.104 + 0.994i)T^{2} \)
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 \( 1 + (-0.669 - 0.743i)T^{2} \)
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 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (-0.669 - 0.743i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
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

−10.23798635397707468950543256969, −9.462975438847406192438140430728, −8.876311868481677839672258240149, −7.922457501152871129786330407692, −7.16774503683869927418924115886, −5.83138800411260401297548223681, −5.11615144442798369969551227209, −3.91891488221525270477614763313, −3.41079661368730100747894974500, −2.07066668989673035872327052712, 0.897428320786826787973931314659, 2.34277045188507119603525639726, 3.68861938348009959087195304882, 4.51060132260213977095644642036, 5.48834226824327514366772477092, 6.51768388583849768727952156672, 7.63017972425821458004172882778, 8.388047248144159511393594420730, 8.809542073501810993817121295644, 9.538626996177761741550904850634

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