Properties

Label 2-11200-1.1-c1-0-13
Degree $2$
Conductor $11200$
Sign $1$
Analytic cond. $89.4324$
Root an. cond. $9.45687$
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  − 3·3-s + 7-s + 6·9-s − 5·11-s + 6·13-s − 17-s − 3·19-s − 3·21-s − 9·27-s + 6·29-s + 4·31-s + 15·33-s − 8·37-s − 18·39-s + 11·41-s − 8·43-s − 2·47-s + 49-s + 3·51-s − 4·53-s + 9·57-s + 4·59-s + 2·61-s + 6·63-s + 9·67-s + 10·71-s − 7·73-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.377·7-s + 2·9-s − 1.50·11-s + 1.66·13-s − 0.242·17-s − 0.688·19-s − 0.654·21-s − 1.73·27-s + 1.11·29-s + 0.718·31-s + 2.61·33-s − 1.31·37-s − 2.88·39-s + 1.71·41-s − 1.21·43-s − 0.291·47-s + 1/7·49-s + 0.420·51-s − 0.549·53-s + 1.19·57-s + 0.520·59-s + 0.256·61-s + 0.755·63-s + 1.09·67-s + 1.18·71-s − 0.819·73-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: $1$
Analytic conductor: \(89.4324\)
Root analytic conductor: \(9.45687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 11200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9169639899\)
\(L(\frac12)\) \(\approx\) \(0.9169639899\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.32920313567782, −16.02448985132609, −15.60094544540022, −15.07206418930860, −13.97129731106014, −13.54476509907924, −12.81294957681673, −12.52324476840778, −11.74476037673387, −11.08219338722428, −10.89933176340543, −10.31307968141347, −9.784053837718050, −8.574488333855631, −8.289899258199497, −7.423322463277878, −6.602299041262052, −6.230222376045967, −5.526910454904173, −4.990596552356908, −4.411020009325725, −3.541236711877093, −2.416047378895859, −1.384753445333907, −0.5256342210606965, 0.5256342210606965, 1.384753445333907, 2.416047378895859, 3.541236711877093, 4.411020009325725, 4.990596552356908, 5.526910454904173, 6.230222376045967, 6.602299041262052, 7.423322463277878, 8.289899258199497, 8.574488333855631, 9.784053837718050, 10.31307968141347, 10.89933176340543, 11.08219338722428, 11.74476037673387, 12.52324476840778, 12.81294957681673, 13.54476509907924, 13.97129731106014, 15.07206418930860, 15.60094544540022, 16.02448985132609, 16.32920313567782

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