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Properties

Label 2-33e2-99.79-c0-0-0
Degree $2$
Conductor $1089$
Sign $-0.460 - 0.887i$
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  + (−1.05 + 0.946i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (1.34 − 0.437i)6-s + (−0.575 − 1.29i)7-s + (0.669 + 0.743i)9-s − 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.82 + 0.813i)14-s + (0.913 − 0.406i)15-s + (0.978 + 0.207i)16-s + (−1.40 − 0.147i)18-s + (0.669 + 0.743i)20-s + 1.41i·21-s + ⋯
L(s)  = 1  + (−1.05 + 0.946i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.669 + 0.743i)5-s + (1.34 − 0.437i)6-s + (−0.575 − 1.29i)7-s + (0.669 + 0.743i)9-s − 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.82 + 0.813i)14-s + (0.913 − 0.406i)15-s + (0.978 + 0.207i)16-s + (−1.40 − 0.147i)18-s + (0.669 + 0.743i)20-s + 1.41i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.460 - 0.887i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2636614731\)
\(L(\frac12)\) \(\approx\) \(0.2636614731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 \)
good2 \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \)
5 \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \)
7 \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \)
13 \( 1 + (-0.913 + 0.406i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.669 - 0.743i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \)
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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

−10.37633005919439845076043536806, −9.564108643939341932419410131596, −8.432555368519922454831476258533, −7.46841669026750260888104470187, −7.14659949396433971246261042802, −6.59863011744108901469845321904, −5.65641789297218858539804508866, −4.30854802329474250784738729025, −3.25997606377375848841490132699, −1.07513108890654075612351258585, 0.47408370384288265223877146384, 2.09084264478343865385506983788, 3.35859812315356422885650446206, 4.51327595723152721171593496064, 5.55057729335510585057097802644, 6.24400564130135902279841143065, 7.63777735757051500568616941818, 8.548253263465825223866503946233, 9.198835227389620170663571675632, 9.735819451222735331028672342442

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