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Properties

Label 2-33e2-3.2-c2-0-40
Degree $2$
Conductor $1089$
Sign $-0.577 - 0.816i$
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  + 3.25i·2-s − 6.59·4-s + 1.59i·5-s + 10.8·7-s − 8.43i·8-s − 5.18·10-s + 9.23·13-s + 35.4i·14-s + 1.07·16-s − 4.81i·17-s + 25.0·19-s − 10.4i·20-s − 36.6i·23-s + 22.4·25-s + 30.0i·26-s + ⋯
L(s)  = 1  + 1.62i·2-s − 1.64·4-s + 0.318i·5-s + 1.55·7-s − 1.05i·8-s − 0.518·10-s + 0.710·13-s + 2.52i·14-s + 0.0670·16-s − 0.283i·17-s + 1.32·19-s − 0.524i·20-s − 1.59i·23-s + 0.898·25-s + 1.15i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.398559468\)
\(L(\frac12)\) \(\approx\) \(2.398559468\)
\(L(2)\) 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 \( 1 - 3.25iT - 4T^{2} \)
5 \( 1 - 1.59iT - 25T^{2} \)
7 \( 1 - 10.8T + 49T^{2} \)
13 \( 1 - 9.23T + 169T^{2} \)
17 \( 1 + 4.81iT - 289T^{2} \)
19 \( 1 - 25.0T + 361T^{2} \)
23 \( 1 + 36.6iT - 529T^{2} \)
29 \( 1 + 24.4iT - 841T^{2} \)
31 \( 1 - 58.1T + 961T^{2} \)
37 \( 1 - 12.7T + 1.36e3T^{2} \)
41 \( 1 - 12.4iT - 1.68e3T^{2} \)
43 \( 1 + 19.6T + 1.84e3T^{2} \)
47 \( 1 - 64.9iT - 2.20e3T^{2} \)
53 \( 1 - 49.5iT - 2.80e3T^{2} \)
59 \( 1 + 24.0iT - 3.48e3T^{2} \)
61 \( 1 + 44.2T + 3.72e3T^{2} \)
67 \( 1 - 33.0T + 4.48e3T^{2} \)
71 \( 1 + 74.9iT - 5.04e3T^{2} \)
73 \( 1 + 41.3T + 5.32e3T^{2} \)
79 \( 1 + 63.9T + 6.24e3T^{2} \)
83 \( 1 + 42.1iT - 6.88e3T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + 108.T + 9.40e3T^{2} \)
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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

−9.680398901379964243358188592442, −8.667099271656398366479043214584, −8.140412362364020184485190765913, −7.53681343372416827071344942438, −6.60591036341371933611884864999, −5.89692627648466230831645638881, −4.85365089395485384487067016413, −4.42930954964403826715762635742, −2.73569867650248035869273539184, −1.04772798670041969001002807062, 1.08172393379005662337233322463, 1.59495915895024368899752192017, 2.91659502106844604145514221631, 3.87921822961421562263597999258, 4.83387666024422936831301611705, 5.47340290711703209952970158899, 7.03696904276948730690431525261, 8.143238244876180746851329379113, 8.701301473621800784439875610510, 9.607983386626305150131568591844

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