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 − 6·11-s − 13-s + 16-s − 3·17-s + 4·19-s + 6·22-s − 3·23-s + 26-s − 3·29-s − 5·31-s − 32-s + 3·34-s + 10·37-s − 4·38-s + 9·41-s + 43-s − 6·44-s + 3·46-s − 52-s + 9·53-s + 3·58-s + 9·59-s − 11·61-s + 5·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.917·19-s + 1.27·22-s − 0.625·23-s + 0.196·26-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 0.514·34-s + 1.64·37-s − 0.648·38-s + 1.40·41-s + 0.152·43-s − 0.904·44-s + 0.442·46-s − 0.138·52-s + 1.23·53-s + 0.393·58-s + 1.17·59-s − 1.40·61-s + 0.635·62-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.85479185296458, −15.47681204276855, −14.81767212795085, −14.27389655476537, −13.54450754443294, −12.97782807184170, −12.68594666091630, −11.86583963155344, −11.18083182028405, −10.96695133243986, −10.13182794161053, −9.844031111749581, −9.162046507577586, −8.554666194643972, −7.894117135360238, −7.456283844116017, −7.074136183766714, −5.930662369944489, −5.721768805811693, −4.889553218425689, −4.193151513610464, −3.252260901953890, −2.510726334924344, −2.082400114822403, −0.8721820604354980, 0, 0.8721820604354980, 2.082400114822403, 2.510726334924344, 3.252260901953890, 4.193151513610464, 4.889553218425689, 5.721768805811693, 5.930662369944489, 7.074136183766714, 7.456283844116017, 7.894117135360238, 8.554666194643972, 9.162046507577586, 9.844031111749581, 10.13182794161053, 10.96695133243986, 11.18083182028405, 11.86583963155344, 12.68594666091630, 12.97782807184170, 13.54450754443294, 14.27389655476537, 14.81767212795085, 15.47681204276855, 15.85479185296458

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