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Properties

Label 2-1053-117.77-c0-0-1
Degree $2$
Conductor $1053$
Sign $-0.984 + 0.173i$
Analytic cond. $0.525515$
Root an. cond. $0.724924$
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  + (0.866 + 1.5i)2-s + (−1 + 1.73i)4-s + (−0.866 + 1.5i)5-s − 1.73·8-s − 3·10-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (−1.73 − 3i)20-s + (−1 − 1.73i)25-s + 1.73·26-s + (1.49 − 2.59i)40-s + (0.5 + 0.866i)43-s + (0.866 + 1.5i)47-s + (−0.5 + 0.866i)49-s + (1.73 − 3i)50-s + ⋯
L(s)  = 1  + (0.866 + 1.5i)2-s + (−1 + 1.73i)4-s + (−0.866 + 1.5i)5-s − 1.73·8-s − 3·10-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (−1.73 − 3i)20-s + (−1 − 1.73i)25-s + 1.73·26-s + (1.49 − 2.59i)40-s + (0.5 + 0.866i)43-s + (0.866 + 1.5i)47-s + (−0.5 + 0.866i)49-s + (1.73 − 3i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(0.525515\)
Root analytic conductor: \(0.724924\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1053,\ (\ :0),\ -0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274589772\)
\(L(\frac12)\) \(\approx\) \(1.274589772\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.77804672645163451567055999713, −9.608552997400485235388681308432, −8.260876305063738486828230124597, −7.84421374709206640889484447594, −7.06962042650512646373723824587, −6.40120070376491987023819865297, −5.67918610040895029947474036861, −4.50844261417428712609051275008, −3.63112894797056687438294681628, −2.88981620915998760789091756648, 0.974709722942862359909312818417, 2.12020921313418719733169410193, 3.62280058907257961823322069057, 4.16421040461620436451638837568, 4.94806068903257175845825198812, 5.72878421458455699466345581077, 7.18049669183365754943293814252, 8.413811449837907192481786018577, 8.981873359459042513195175512690, 9.812033177476301740131527037243

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