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 + 2·13-s + 16-s + 4·17-s − 20-s + 2·22-s − 8·23-s + 25-s − 2·26-s − 2·31-s − 32-s − 4·34-s + 8·37-s + 40-s + 2·41-s − 2·43-s − 2·44-s + 8·46-s − 10·47-s − 50-s + 2·52-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 + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.223·20-s + 0.426·22-s − 1.66·23-s + 1/5·25-s − 0.392·26-s − 0.359·31-s − 0.176·32-s − 0.685·34-s + 1.31·37-s + 0.158·40-s + 0.312·41-s − 0.304·43-s − 0.301·44-s + 1.17·46-s − 1.45·47-s − 0.141·50-s + 0.277·52-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−18.25692077033233, −17.93843035358430, −16.79055155786647, −16.61061660838142, −15.83683108205686, −15.44885733548015, −14.64218328819377, −14.05799968561976, −13.27669202138309, −12.50828822474366, −12.01244651779843, −11.23102695229275, −10.76865740195623, −9.907448664363188, −9.555839307443521, −8.530120923228090, −7.950071907738147, −7.629563066648573, −6.573230364735012, −5.947050546028574, −5.120931894665405, −4.066295261972232, −3.310871635914721, −2.332047982812331, −1.249797139061019, 0, 1.249797139061019, 2.332047982812331, 3.310871635914721, 4.066295261972232, 5.120931894665405, 5.947050546028574, 6.573230364735012, 7.629563066648573, 7.950071907738147, 8.530120923228090, 9.555839307443521, 9.907448664363188, 10.76865740195623, 11.23102695229275, 12.01244651779843, 12.50828822474366, 13.27669202138309, 14.05799968561976, 14.64218328819377, 15.44885733548015, 15.83683108205686, 16.61061660838142, 16.79055155786647, 17.93843035358430, 18.25692077033233

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