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 + 2·13-s + 16-s + 6·17-s + 4·19-s + 2·26-s + 6·29-s + 4·31-s + 32-s + 6·34-s − 2·37-s + 4·38-s + 6·41-s − 8·43-s + 12·47-s + 2·52-s + 6·53-s + 6·58-s − 12·59-s − 2·61-s + 4·62-s + 64-s − 8·67-s + 6·68-s + 14·73-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.392·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 0.277·52-s + 0.824·53-s + 0.787·58-s − 1.56·59-s − 0.256·61-s + 0.508·62-s + 1/8·64-s − 0.977·67-s + 0.727·68-s + 1.63·73-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

\(L(1)\) \(\approx\) \(4.661893839\)
\(L(\frac12)\) \(\approx\) \(4.661893839\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−15.46499294913583, −15.05120056401999, −14.17884847388272, −13.99942240040866, −13.55050988175178, −12.73370562760139, −12.30839344374932, −11.80729462328842, −11.35859228986428, −10.50228703075689, −10.20809001761956, −9.533528580086283, −8.789723611575302, −8.186383612687196, −7.523448990410824, −7.100575804126287, −6.184374175484243, −5.851249878542204, −5.138766389542204, −4.545366851212120, −3.768116576894883, −3.164514431651817, −2.601007568462360, −1.486652609059844, −0.8513267056812171, 0.8513267056812171, 1.486652609059844, 2.601007568462360, 3.164514431651817, 3.768116576894883, 4.545366851212120, 5.138766389542204, 5.851249878542204, 6.184374175484243, 7.100575804126287, 7.523448990410824, 8.186383612687196, 8.789723611575302, 9.533528580086283, 10.20809001761956, 10.50228703075689, 11.35859228986428, 11.80729462328842, 12.30839344374932, 12.73370562760139, 13.55050988175178, 13.99942240040866, 14.17884847388272, 15.05120056401999, 15.46499294913583

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