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 − 6·13-s + 16-s − 2·17-s − 4·22-s − 6·26-s − 6·29-s − 8·31-s + 32-s − 2·34-s + 10·37-s + 2·41-s − 4·43-s − 4·44-s − 8·47-s − 6·52-s − 2·53-s − 6·58-s − 8·59-s + 14·61-s − 8·62-s + 64-s + 12·67-s − 2·68-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.852·22-s − 1.17·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.603·44-s − 1.16·47-s − 0.832·52-s − 0.274·53-s − 0.787·58-s − 1.04·59-s + 1.79·61-s − 1.01·62-s + 1/8·64-s + 1.46·67-s − 0.242·68-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

\(L(1)\) \(\approx\) \(1.842974556\)
\(L(\frac12)\) \(\approx\) \(1.842974556\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−15.38716780641268, −14.97037922487492, −14.44324528167688, −14.04492975802030, −13.12105133167618, −12.76457433077132, −12.68884377529301, −11.56976781543707, −11.34836723138370, −10.71343058538970, −9.944378908161039, −9.659649053989728, −8.891804338028523, −7.941304274751658, −7.678926491648713, −7.044336333131375, −6.419710590317103, −5.558200675576844, −5.162785822598946, −4.636085217564663, −3.853396567982809, −3.089607628972951, −2.352025437728617, −1.936781269115221, −0.4532593565291576, 0.4532593565291576, 1.936781269115221, 2.352025437728617, 3.089607628972951, 3.853396567982809, 4.636085217564663, 5.162785822598946, 5.558200675576844, 6.419710590317103, 7.044336333131375, 7.678926491648713, 7.941304274751658, 8.891804338028523, 9.659649053989728, 9.944378908161039, 10.71343058538970, 11.34836723138370, 11.56976781543707, 12.68884377529301, 12.76457433077132, 13.12105133167618, 14.04492975802030, 14.44324528167688, 14.97037922487492, 15.38716780641268

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