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

\(L(1)\) \(\approx\) \(0.7464900134\)
\(L(\frac12)\) \(\approx\) \(0.7464900134\)
\(L(\frac{3}{2})\) 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

−16.40542542441847, −15.97704291584822, −15.21670897994028, −14.73998242077803, −14.35430593213636, −13.56205134854206, −13.13489153551473, −12.40256814714031, −11.81217258931129, −11.38905719204158, −10.58374981293480, −10.17029059608129, −9.476457864892039, −8.725303103552751, −8.211680748931110, −7.548356010711300, −7.059548177743493, −6.146158259854858, −5.355148839461707, −5.026980942147259, −4.260516779141184, −3.140007055055890, −2.572459115633244, −1.904547143488513, −0.3708024518635366, 0.3708024518635366, 1.904547143488513, 2.572459115633244, 3.140007055055890, 4.260516779141184, 5.026980942147259, 5.355148839461707, 6.146158259854858, 7.059548177743493, 7.548356010711300, 8.211680748931110, 8.725303103552751, 9.476457864892039, 10.17029059608129, 10.58374981293480, 11.38905719204158, 11.81217258931129, 12.40256814714031, 13.13489153551473, 13.56205134854206, 14.35430593213636, 14.73998242077803, 15.21670897994028, 15.97704291584822, 16.40542542441847

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