Properties

Label 2-22050-1.1-c1-0-81
Degree $2$
Conductor $22050$
Sign $1$
Analytic cond. $176.070$
Root an. cond. $13.2691$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 3·13-s + 16-s + 7·17-s + 6·19-s + 4·22-s − 9·23-s + 3·26-s + 3·29-s + 7·31-s + 32-s + 7·34-s − 10·37-s + 6·38-s + 41-s + 13·43-s + 4·44-s − 9·46-s − 2·47-s + 3·52-s + 53-s + 3·58-s + 11·59-s − 13·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s + 0.832·13-s + 1/4·16-s + 1.69·17-s + 1.37·19-s + 0.852·22-s − 1.87·23-s + 0.588·26-s + 0.557·29-s + 1.25·31-s + 0.176·32-s + 1.20·34-s − 1.64·37-s + 0.973·38-s + 0.156·41-s + 1.98·43-s + 0.603·44-s − 1.32·46-s − 0.291·47-s + 0.416·52-s + 0.137·53-s + 0.393·58-s + 1.43·59-s − 1.66·61-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: $1$
Analytic conductor: \(176.070\)
Root analytic conductor: \(13.2691\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{22050} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 22050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.217350467\)
\(L(\frac12)\) \(\approx\) \(5.217350467\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.73994278121463, −14.83492112176473, −14.14459477264117, −14.05895188656362, −13.65826263828132, −12.72248430572725, −12.08069423142824, −11.92416476647649, −11.47021321450050, −10.43068332949468, −10.23602357416993, −9.448412678089143, −8.931234924277081, −7.995496002442261, −7.765285647306731, −6.908092188612745, −6.273804282549111, −5.822595845287179, −5.240440547868123, −4.402453677935162, −3.687583063311511, −3.416392263470864, −2.466070258220541, −1.449176806587221, −0.9357648134445053, 0.9357648134445053, 1.449176806587221, 2.466070258220541, 3.416392263470864, 3.687583063311511, 4.402453677935162, 5.240440547868123, 5.822595845287179, 6.273804282549111, 6.908092188612745, 7.765285647306731, 7.995496002442261, 8.931234924277081, 9.448412678089143, 10.23602357416993, 10.43068332949468, 11.47021321450050, 11.92416476647649, 12.08069423142824, 12.72248430572725, 13.65826263828132, 14.05895188656362, 14.14459477264117, 14.83492112176473, 15.73994278121463

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