Properties

Degree $2$
Conductor $22050$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 3·11-s + 4·13-s + 16-s − 4·19-s + 3·22-s − 4·26-s − 9·29-s − 31-s − 32-s − 8·37-s + 4·38-s + 10·43-s − 3·44-s − 6·47-s + 4·52-s − 3·53-s + 9·58-s − 3·59-s − 10·61-s + 62-s + 64-s + 10·67-s + 6·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s + 1.10·13-s + 1/4·16-s − 0.917·19-s + 0.639·22-s − 0.784·26-s − 1.67·29-s − 0.179·31-s − 0.176·32-s − 1.31·37-s + 0.648·38-s + 1.52·43-s − 0.452·44-s − 0.875·47-s + 0.554·52-s − 0.412·53-s + 1.18·58-s − 0.390·59-s − 1.28·61-s + 0.127·62-s + 1/8·64-s + 1.22·67-s + 0.712·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22050\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{22050} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9144433491\)
\(L(\frac12)\) \(\approx\) \(0.9144433491\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.67429309096918, −15.18835828032219, −14.52542155605247, −13.90799553913967, −13.31019358467246, −12.71242245105317, −12.42477381191592, −11.35968592912173, −11.14282023331823, −10.57446771550605, −10.10248728299943, −9.298806121991184, −8.903013131116436, −8.262056391559943, −7.786401198057315, −7.169037953753891, −6.493497749426056, −5.852735287540278, −5.360265201277213, −4.449782065409162, −3.696294192091178, −3.053647249524689, −2.133288516008464, −1.564783322508771, −0.4270440021900709, 0.4270440021900709, 1.564783322508771, 2.133288516008464, 3.053647249524689, 3.696294192091178, 4.449782065409162, 5.360265201277213, 5.852735287540278, 6.493497749426056, 7.169037953753891, 7.786401198057315, 8.262056391559943, 8.903013131116436, 9.298806121991184, 10.10248728299943, 10.57446771550605, 11.14282023331823, 11.35968592912173, 12.42477381191592, 12.71242245105317, 13.31019358467246, 13.90799553913967, 14.52542155605247, 15.18835828032219, 15.67429309096918

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