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 − 3·11-s − 2·13-s + 16-s − 6·17-s + 2·19-s + 3·22-s + 6·23-s + 2·26-s + 9·29-s − 7·31-s − 32-s + 6·34-s + 10·37-s − 2·38-s + 4·43-s − 3·44-s − 6·46-s − 12·47-s − 2·52-s + 3·53-s − 9·58-s − 3·59-s − 4·61-s + 7·62-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 − 1.45·17-s + 0.458·19-s + 0.639·22-s + 1.25·23-s + 0.392·26-s + 1.67·29-s − 1.25·31-s − 0.176·32-s + 1.02·34-s + 1.64·37-s − 0.324·38-s + 0.609·43-s − 0.452·44-s − 0.884·46-s − 1.75·47-s − 0.277·52-s + 0.412·53-s − 1.18·58-s − 0.390·59-s − 0.512·61-s + 0.889·62-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.71734497257021, −15.49344869936259, −14.78872587659237, −14.33833444329485, −13.53095953867940, −12.96603717055081, −12.69888577894459, −11.80970263686011, −11.30332370016090, −10.85422629384332, −10.28158330164900, −9.719209777471031, −9.061940566320575, −8.711143371499364, −7.875170321930661, −7.531334429005399, −6.783877890953273, −6.350072048048862, −5.468772028905383, −4.848616133019589, −4.297487935571765, −3.132331091289517, −2.688351797358801, −1.947814853447014, −0.9277367512032092, 0, 0.9277367512032092, 1.947814853447014, 2.688351797358801, 3.132331091289517, 4.297487935571765, 4.848616133019589, 5.468772028905383, 6.350072048048862, 6.783877890953273, 7.531334429005399, 7.875170321930661, 8.711143371499364, 9.061940566320575, 9.719209777471031, 10.28158330164900, 10.85422629384332, 11.30332370016090, 11.80970263686011, 12.69888577894459, 12.96603717055081, 13.53095953867940, 14.33833444329485, 14.78872587659237, 15.49344869936259, 15.71734497257021

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