Properties

Degree $2$
Conductor $11200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 3·9-s + 4·11-s − 6·13-s − 2·17-s − 6·29-s − 8·31-s − 10·37-s + 2·41-s − 4·43-s + 8·47-s + 49-s − 2·53-s − 8·59-s + 14·61-s + 3·63-s + 12·67-s + 16·71-s − 2·73-s − 4·77-s + 8·79-s + 9·81-s − 8·83-s + 10·89-s + 6·91-s − 2·97-s − 12·99-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s + 1.20·11-s − 1.66·13-s − 0.485·17-s − 1.11·29-s − 1.43·31-s − 1.64·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.04·59-s + 1.79·61-s + 0.377·63-s + 1.46·67-s + 1.89·71-s − 0.234·73-s − 0.455·77-s + 0.900·79-s + 81-s − 0.878·83-s + 1.05·89-s + 0.628·91-s − 0.203·97-s − 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11200\)    =    \(2^{6} \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{11200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9912922790\)
\(L(\frac12)\) \(\approx\) \(0.9912922790\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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

−16.73372781801255, −15.88930639178593, −15.27684321780343, −14.56306399657711, −14.39782157083042, −13.76416291022962, −12.94318945912598, −12.38840992089542, −11.92980874993171, −11.30404182037826, −10.78275699823388, −9.951186260354586, −9.298277293803813, −9.047539951711544, −8.251611232435457, −7.416612639710277, −6.922528072022910, −6.307683576973520, −5.415872288776436, −5.049059885035755, −3.938994409757820, −3.482844279846286, −2.466261182628301, −1.862680866993682, −0.4355700756094492, 0.4355700756094492, 1.862680866993682, 2.466261182628301, 3.482844279846286, 3.938994409757820, 5.049059885035755, 5.415872288776436, 6.307683576973520, 6.922528072022910, 7.416612639710277, 8.251611232435457, 9.047539951711544, 9.298277293803813, 9.951186260354586, 10.78275699823388, 11.30404182037826, 11.92980874993171, 12.38840992089542, 12.94318945912598, 13.76416291022962, 14.39782157083042, 14.56306399657711, 15.27684321780343, 15.88930639178593, 16.73372781801255

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