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Properties

Label 2-33e2-11.10-c2-0-24
Degree $2$
Conductor $1089$
Sign $-0.522 - 0.852i$
Analytic cond. $29.6731$
Root an. cond. $5.44730$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 13.9i·7-s + 25.3i·13-s + 16·16-s − 17.9i·19-s − 25·25-s + 55.6i·28-s − 19.0·31-s − 57.1·37-s − 43.2i·43-s − 144.·49-s + 101. i·52-s + 74.5i·61-s + 64·64-s + 133.·67-s + ⋯
L(s)  = 1  + 4-s + 1.98i·7-s + 1.95i·13-s + 16-s − 0.944i·19-s − 25-s + 1.98i·28-s − 0.614·31-s − 1.54·37-s − 1.00i·43-s − 2.94·49-s + 1.95i·52-s + 1.22i·61-s + 64-s + 1.99·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ -0.522 - 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.008005831\)
\(L(\frac12)\) \(\approx\) \(2.008005831\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 - 4T^{2} \)
5 \( 1 + 25T^{2} \)
7 \( 1 - 13.9iT - 49T^{2} \)
13 \( 1 - 25.3iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 19.0T + 961T^{2} \)
37 \( 1 + 57.1T + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 43.2iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 74.5iT - 3.72e3T^{2} \)
67 \( 1 - 133.T + 4.48e3T^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + 80.2iT - 5.32e3T^{2} \)
79 \( 1 - 119. iT - 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 - 169T + 9.40e3T^{2} \)
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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

−9.802409044540315125853223400007, −9.043325544390322411041945911402, −8.517073997269687695359082705128, −7.27724205087994399926813826175, −6.57010707662717562112229433224, −5.81391257831634281237744013619, −4.96867093587695650441475715271, −3.57195405562074309341235903271, −2.31788027543056941841267601759, −1.90397316010234427017649175003, 0.55690932224552087863671425684, 1.69141229278673848980375572760, 3.23724055551118765324397593151, 3.79515142242861088262579994126, 5.16751639370898943043530766928, 6.11088014947879169737949370208, 7.01526325845226768561463018702, 7.74471131693019208966964451970, 8.113068159815200511199648716324, 9.812912708524219927067250856327

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