Properties

Label 2-33e2-99.61-c0-0-1
Degree $2$
Conductor $1089$
Sign $0.736 - 0.676i$
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  + (−0.294 + 1.38i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)5-s + (1.34 + 0.437i)6-s + (1.40 − 0.147i)7-s + (−0.978 − 0.207i)9-s + 1.41i·10-s + (−0.499 + 0.866i)12-s + (−0.209 + 1.98i)14-s + (−0.104 − 0.994i)15-s + (−0.669 − 0.743i)16-s + (0.575 − 1.29i)18-s + (−0.978 − 0.207i)20-s − 1.41i·21-s + ⋯
L(s)  = 1  + (−0.294 + 1.38i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)5-s + (1.34 + 0.437i)6-s + (1.40 − 0.147i)7-s + (−0.978 − 0.207i)9-s + 1.41i·10-s + (−0.499 + 0.866i)12-s + (−0.209 + 1.98i)14-s + (−0.104 − 0.994i)15-s + (−0.669 − 0.743i)16-s + (0.575 − 1.29i)18-s + (−0.978 − 0.207i)20-s − 1.41i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.736 - 0.676i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158160485\)
\(L(\frac12)\) \(\approx\) \(1.158160485\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 \)
good2 \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \)
5 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
7 \( 1 + (-1.40 + 0.147i)T + (0.978 - 0.207i)T^{2} \)
13 \( 1 + (0.104 + 0.994i)T^{2} \)
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 \( 1 + (1.40 - 0.147i)T + (0.978 - 0.207i)T^{2} \)
31 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
61 \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \)
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 \( 1 + (-0.913 - 0.406i)T^{2} \)
83 \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \)
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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

−9.780913938632627789123307555218, −8.954839888189361746274498112162, −8.273070012858916757063266056548, −7.60948092129893710968710155010, −7.01153041783487104103839633501, −5.83006522259742373040152351296, −5.65487009880439745244214764988, −4.47850130754752118890978442312, −2.50810393779430471344256421674, −1.48115754420001672506496018547, 1.65494580223565879473431541364, 2.45605676292246374281323341074, 3.54415247109377672229603464551, 4.58717201815765784123149877818, 5.37349177261154745929041675063, 6.39810963985813889872544125275, 7.936408522708452807586447006549, 8.713233266948904268790719432931, 9.496785860998779115992817420606, 10.05261725663487092780729080077

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