Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 6·11-s + 4·13-s + 16-s − 3·17-s + 4·19-s + 6·22-s − 3·23-s − 4·26-s + 6·29-s − 5·31-s − 32-s + 3·34-s + 8·37-s − 4·38-s − 3·41-s + 8·43-s − 6·44-s + 3·46-s + 9·47-s + 4·52-s − 12·53-s − 6·58-s + 6·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 1.10·13-s + 1/4·16-s − 0.727·17-s + 0.917·19-s + 1.27·22-s − 0.625·23-s − 0.784·26-s + 1.11·29-s − 0.898·31-s − 0.176·32-s + 0.514·34-s + 1.31·37-s − 0.648·38-s − 0.468·41-s + 1.21·43-s − 0.904·44-s + 0.442·46-s + 1.31·47-s + 0.554·52-s − 1.64·53-s − 0.787·58-s + 0.781·59-s − 0.256·61-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

\(L(1)\) \(\approx\) \(1.139971902\)
\(L(\frac12)\) \(\approx\) \(1.139971902\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.69022613800185, −15.35729057255893, −14.36794697229817, −13.95644347538289, −13.25363746618900, −12.88847937071214, −12.28082731984810, −11.46151769389581, −11.07394132640262, −10.56683045079804, −10.06125046282913, −9.444513017067354, −8.793663774662732, −8.258260420242863, −7.742486586042291, −7.279031798512125, −6.443374403356086, −5.855923899832222, −5.319141089671593, −4.515865977526442, −3.728533400610443, −2.837995545587596, −2.405813249687509, −1.404174750285233, −0.4989414208719539, 0.4989414208719539, 1.404174750285233, 2.405813249687509, 2.837995545587596, 3.728533400610443, 4.515865977526442, 5.319141089671593, 5.855923899832222, 6.443374403356086, 7.279031798512125, 7.742486586042291, 8.258260420242863, 8.793663774662732, 9.444513017067354, 10.06125046282913, 10.56683045079804, 11.07394132640262, 11.46151769389581, 12.28082731984810, 12.88847937071214, 13.25363746618900, 13.95644347538289, 14.36794697229817, 15.35729057255893, 15.69022613800185

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