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 − 2·11-s + 2·13-s + 16-s + 4·17-s − 2·22-s + 8·23-s + 2·26-s + 2·31-s + 32-s + 4·34-s − 8·37-s − 2·41-s + 2·43-s − 2·44-s + 8·46-s − 10·47-s + 2·52-s − 2·53-s + 4·59-s + 10·61-s + 2·62-s + 64-s − 2·67-s + 4·68-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 0.554·13-s + 1/4·16-s + 0.970·17-s − 0.426·22-s + 1.66·23-s + 0.392·26-s + 0.359·31-s + 0.176·32-s + 0.685·34-s − 1.31·37-s − 0.312·41-s + 0.304·43-s − 0.301·44-s + 1.17·46-s − 1.45·47-s + 0.277·52-s − 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.254·62-s + 1/8·64-s − 0.244·67-s + 0.485·68-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

\(L(1)\) \(\approx\) \(3.983339330\)
\(L(\frac12)\) \(\approx\) \(3.983339330\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.43390909488395, −14.96296644867933, −14.47095636474819, −13.85253587819716, −13.35653948466609, −12.89966955868024, −12.33979900363166, −11.84099711773001, −11.04638245106021, −10.84871116092659, −10.06695724959743, −9.562147495507480, −8.762459558171668, −8.164721005802471, −7.667667995161204, −6.766863353303322, −6.604679636303829, −5.471693128799512, −5.309387542515321, −4.608252536994640, −3.659936193938271, −3.266138290196486, −2.515292461403804, −1.605035114760778, −0.7433083758122876, 0.7433083758122876, 1.605035114760778, 2.515292461403804, 3.266138290196486, 3.659936193938271, 4.608252536994640, 5.309387542515321, 5.471693128799512, 6.604679636303829, 6.766863353303322, 7.667667995161204, 8.164721005802471, 8.762459558171668, 9.562147495507480, 10.06695724959743, 10.84871116092659, 11.04638245106021, 11.84099711773001, 12.33979900363166, 12.89966955868024, 13.35653948466609, 13.85253587819716, 14.47095636474819, 14.96296644867933, 15.43390909488395

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