Properties

Degree $2$
Conductor $4410$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s + 2·11-s + 16-s + 4·17-s − 6·19-s + 20-s − 2·22-s − 3·23-s + 25-s − 9·29-s − 4·31-s − 32-s − 4·34-s − 4·37-s + 6·38-s − 40-s + 7·41-s − 5·43-s + 2·44-s + 3·46-s − 8·47-s − 50-s + 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.603·11-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 1.67·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.657·37-s + 0.973·38-s − 0.158·40-s + 1.09·41-s − 0.762·43-s + 0.301·44-s + 0.442·46-s − 1.16·47-s − 0.141·50-s + 0.274·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{4410} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−18.23195452534123, −17.64731433231838, −16.98401016676670, −16.62063863253663, −16.07383583170256, −15.01069446577625, −14.79215140885913, −14.07117759988401, −13.24185654603247, −12.62616324559331, −11.99471194174597, −11.25863350521189, −10.62209062005163, −10.03444865410556, −9.296976854386146, −8.879081814003998, −7.978867865315999, −7.430850853630142, −6.512156532796291, −5.998696488601452, −5.184030010498902, −4.082262668205959, −3.307793128582229, −2.133418181014948, −1.464500585647806, 0, 1.464500585647806, 2.133418181014948, 3.307793128582229, 4.082262668205959, 5.184030010498902, 5.998696488601452, 6.512156532796291, 7.430850853630142, 7.978867865315999, 8.879081814003998, 9.296976854386146, 10.03444865410556, 10.62209062005163, 11.25863350521189, 11.99471194174597, 12.62616324559331, 13.24185654603247, 14.07117759988401, 14.79215140885913, 15.01069446577625, 16.07383583170256, 16.62063863253663, 16.98401016676670, 17.64731433231838, 18.23195452534123

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