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 − 2·13-s + 16-s + 3·17-s + 7·19-s + 3·22-s + 2·26-s + 6·29-s + 4·31-s − 32-s − 3·34-s + 8·37-s − 7·38-s − 9·41-s + 8·43-s − 3·44-s − 6·47-s − 2·52-s + 12·53-s − 6·58-s + 12·59-s + 10·61-s − 4·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s − 0.554·13-s + 1/4·16-s + 0.727·17-s + 1.60·19-s + 0.639·22-s + 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 1.31·37-s − 1.13·38-s − 1.40·41-s + 1.21·43-s − 0.452·44-s − 0.875·47-s − 0.277·52-s + 1.64·53-s − 0.787·58-s + 1.56·59-s + 1.28·61-s − 0.508·62-s + 1/8·64-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

\(L(1)\) \(\approx\) \(1.573250152\)
\(L(\frac12)\) \(\approx\) \(1.573250152\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.64189293208599, −15.08363902048432, −14.54553121385849, −13.91069442701851, −13.39266740668764, −12.77190711971305, −12.07185861557012, −11.72200549419279, −11.17865485767944, −10.28392825626033, −10.02940817296462, −9.640834611301561, −8.772278834099148, −8.259848080579670, −7.683542218873003, −7.242100271096504, −6.605275807537992, −5.727484980726217, −5.303684342074800, −4.610315145984566, −3.658621859228769, −2.832815335123097, −2.473698234812082, −1.295716646638888, −0.6238007583305811, 0.6238007583305811, 1.295716646638888, 2.473698234812082, 2.832815335123097, 3.658621859228769, 4.610315145984566, 5.303684342074800, 5.727484980726217, 6.605275807537992, 7.242100271096504, 7.683542218873003, 8.259848080579670, 8.772278834099148, 9.640834611301561, 10.02940817296462, 10.28392825626033, 11.17865485767944, 11.72200549419279, 12.07185861557012, 12.77190711971305, 13.39266740668764, 13.91069442701851, 14.54553121385849, 15.08363902048432, 15.64189293208599

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