Properties

Label 2-33e2-99.43-c0-0-0
Degree $2$
Conductor $1089$
Sign $0.576 - 0.816i$
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  + (−1.22 − 0.707i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 1.41i·6-s + (1.22 + 0.707i)7-s + (−0.499 + 0.866i)9-s − 1.41i·10-s + (−0.5 + 0.866i)12-s + (−0.999 − 1.73i)14-s + (−0.499 + 0.866i)15-s + (0.499 − 0.866i)16-s + (1.22 − 0.707i)18-s + (−0.5 + 0.866i)20-s + 1.41i·21-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 1.41i·6-s + (1.22 + 0.707i)7-s + (−0.499 + 0.866i)9-s − 1.41i·10-s + (−0.5 + 0.866i)12-s + (−0.999 − 1.73i)14-s + (−0.499 + 0.866i)15-s + (0.499 − 0.866i)16-s + (1.22 − 0.707i)18-s + (−0.5 + 0.866i)20-s + 1.41i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.576 - 0.816i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.576 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7765913552\)
\(L(\frac12)\) \(\approx\) \(0.7765913552\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 \)
good2 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.18996267546063079124524750598, −9.417545586459181508933074132583, −8.776004512457878264253289523690, −8.103260110408192335334800528180, −7.31867360494467816568380471352, −5.79353587906435914021904395580, −5.04676289425161766849602400909, −3.70689898853406302105373239916, −2.45344756700752563808013222718, −1.98457268831581007194767202521, 1.11762132914441298332085719015, 1.83265225700011097594630289268, 3.68889216417039703106138956221, 5.01557043382854351366197750403, 5.95632418316677480908896963779, 7.16771593774900244636901159438, 7.43594362601924423811822942129, 8.440264848888841961529340737222, 8.782003164320797828501290474608, 9.535163671635359309791411885925

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