Properties

Degree $2$
Conductor $2070$
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 − 4·11-s − 2·13-s + 16-s + 6·17-s + 4·19-s − 20-s + 4·22-s + 23-s + 25-s + 2·26-s + 2·29-s − 32-s − 6·34-s − 2·37-s − 4·38-s + 40-s − 10·41-s − 4·43-s − 4·44-s − 46-s − 7·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.371·29-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.609·43-s − 0.603·44-s − 0.147·46-s − 49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2070} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.50234704852203, −18.98874005090660, −18.31281649445369, −17.95507409746247, −16.89017558815082, −16.60644510575062, −15.73149512181039, −15.35111424420008, −14.50944947423345, −13.80804465836127, −12.90038668611946, −12.19960000816225, −11.66800856721595, −10.80691166017936, −10.09137212961524, −9.664589981070879, −8.609088489832512, −7.877013836580382, −7.495811251421517, −6.579979843118159, −5.448411665587810, −4.900369829758244, −3.442210252179751, −2.790626623842764, −1.415434150210406, 0, 1.415434150210406, 2.790626623842764, 3.442210252179751, 4.900369829758244, 5.448411665587810, 6.579979843118159, 7.495811251421517, 7.877013836580382, 8.609088489832512, 9.664589981070879, 10.09137212961524, 10.80691166017936, 11.66800856721595, 12.19960000816225, 12.90038668611946, 13.80804465836127, 14.50944947423345, 15.35111424420008, 15.73149512181039, 16.60644510575062, 16.89017558815082, 17.95507409746247, 18.31281649445369, 18.98874005090660, 19.50234704852203

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