Properties

Degree $2$
Conductor $22050$
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 − 8-s − 2·11-s + 13-s + 16-s + 3·17-s + 2·22-s + 23-s − 26-s + 5·29-s − 7·31-s − 32-s − 3·34-s − 2·37-s + 7·41-s − 11·43-s − 2·44-s − 46-s + 8·47-s + 52-s + 53-s − 5·58-s − 5·59-s + 3·61-s + 7·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.603·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.426·22-s + 0.208·23-s − 0.196·26-s + 0.928·29-s − 1.25·31-s − 0.176·32-s − 0.514·34-s − 0.328·37-s + 1.09·41-s − 1.67·43-s − 0.301·44-s − 0.147·46-s + 1.16·47-s + 0.138·52-s + 0.137·53-s − 0.656·58-s − 0.650·59-s + 0.384·61-s + 0.889·62-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: \(1\)
Selberg data: \((2,\ 22050,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.92080778171648, −15.28254144361946, −14.92539684855713, −14.11332999613497, −13.79368755288236, −12.82808252925826, −12.69027641566000, −11.79671493868903, −11.47022499730125, −10.62309551714845, −10.37501616813550, −9.790055059041796, −8.973916338745140, −8.743219320953742, −7.852733777503376, −7.565999396284116, −6.865327833606074, −6.172494811182432, −5.563696152175834, −4.958128431223325, −4.086184918049055, −3.288446746105958, −2.695077347689118, −1.811031183949682, −1.027535277117364, 0, 1.027535277117364, 1.811031183949682, 2.695077347689118, 3.288446746105958, 4.086184918049055, 4.958128431223325, 5.563696152175834, 6.172494811182432, 6.865327833606074, 7.565999396284116, 7.852733777503376, 8.743219320953742, 8.973916338745140, 9.790055059041796, 10.37501616813550, 10.62309551714845, 11.47022499730125, 11.79671493868903, 12.69027641566000, 12.82808252925826, 13.79368755288236, 14.11332999613497, 14.92539684855713, 15.28254144361946, 15.92080778171648

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