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 − 2·13-s + 16-s − 6·17-s + 4·19-s − 20-s + 25-s + 2·26-s + 6·29-s + 4·31-s − 32-s + 6·34-s + 2·37-s − 4·38-s + 40-s + 6·41-s + 8·43-s − 12·47-s − 50-s − 2·52-s − 6·53-s − 6·58-s − 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.21·43-s − 1.75·47-s − 0.141·50-s − 0.277·52-s − 0.824·53-s − 0.787·58-s − 1.56·59-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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−18.18915534258719, −17.51222333510202, −17.36800025204587, −16.23658895590996, −15.97671736408131, −15.40567908733436, −14.67116720708100, −14.04282686807304, −13.28334080073009, −12.53964868467596, −11.93135076940038, −11.28298145750904, −10.78872341215883, −9.951149409349320, −9.380602186444961, −8.714911752814454, −7.991006296285731, −7.418803344007690, −6.665251587209607, −6.033476394797635, −4.888230967700311, −4.319772027611907, −3.125905420974737, −2.433101529325110, −1.220610903438796, 0, 1.220610903438796, 2.433101529325110, 3.125905420974737, 4.319772027611907, 4.888230967700311, 6.033476394797635, 6.665251587209607, 7.418803344007690, 7.991006296285731, 8.714911752814454, 9.380602186444961, 9.951149409349320, 10.78872341215883, 11.28298145750904, 11.93135076940038, 12.53964868467596, 13.28334080073009, 14.04282686807304, 14.67116720708100, 15.40567908733436, 15.97671736408131, 16.23658895590996, 17.36800025204587, 17.51222333510202, 18.18915534258719

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