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 − 4·11-s − 2·13-s + 16-s − 2·17-s + 4·19-s − 4·22-s − 8·23-s − 2·26-s − 6·29-s + 8·31-s + 32-s − 2·34-s + 2·37-s + 4·38-s + 2·41-s + 12·43-s − 4·44-s − 8·46-s + 8·47-s − 2·52-s + 6·53-s − 6·58-s + 4·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.852·22-s − 1.66·23-s − 0.392·26-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.328·37-s + 0.648·38-s + 0.312·41-s + 1.82·43-s − 0.603·44-s − 1.17·46-s + 1.16·47-s − 0.277·52-s + 0.824·53-s − 0.787·58-s + 0.520·59-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

\(L(1)\) \(\approx\) \(2.619132225\)
\(L(\frac12)\) \(\approx\) \(2.619132225\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−15.67989661381016, −14.91351847524405, −14.44158266091788, −13.82853517901885, −13.39441484015432, −12.94208178518241, −12.23307907604422, −11.84467735035530, −11.31341743305836, −10.44990519970956, −10.27536948514556, −9.523935577364385, −8.860245399927747, −8.045448413776216, −7.537469875028804, −7.205414615665639, −6.194230042969825, −5.720189807141514, −5.263285496932080, −4.316199986806723, −4.104079401653778, −2.918812252403384, −2.585143861569479, −1.769331835916199, −0.5601283777211909, 0.5601283777211909, 1.769331835916199, 2.585143861569479, 2.918812252403384, 4.104079401653778, 4.316199986806723, 5.263285496932080, 5.720189807141514, 6.194230042969825, 7.205414615665639, 7.537469875028804, 8.045448413776216, 8.860245399927747, 9.523935577364385, 10.27536948514556, 10.44990519970956, 11.31341743305836, 11.84467735035530, 12.23307907604422, 12.94208178518241, 13.39441484015432, 13.82853517901885, 14.44158266091788, 14.91351847524405, 15.67989661381016

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