Properties

Label 2-11200-1.1-c1-0-56
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s − 2·9-s − 3·11-s + 5·13-s − 3·17-s + 2·19-s − 21-s − 6·23-s + 5·27-s − 3·29-s + 4·31-s + 3·33-s + 2·37-s − 5·39-s − 12·41-s + 10·43-s + 9·47-s + 49-s + 3·51-s + 12·53-s − 2·57-s − 8·61-s − 2·63-s + 4·67-s + 6·69-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.38·13-s − 0.727·17-s + 0.458·19-s − 0.218·21-s − 1.25·23-s + 0.962·27-s − 0.557·29-s + 0.718·31-s + 0.522·33-s + 0.328·37-s − 0.800·39-s − 1.87·41-s + 1.52·43-s + 1.31·47-s + 1/7·49-s + 0.420·51-s + 1.64·53-s − 0.264·57-s − 1.02·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.234·73-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: \(1\)
Selberg data: \((2,\ 11200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−16.82568913563879, −16.14178141825553, −15.62585477991060, −15.31092016975847, −14.37355364337805, −13.78049715599735, −13.49163247014587, −12.74925979699793, −11.93754075419907, −11.65109809558644, −10.86422319574373, −10.64193034879863, −9.894871311776432, −8.987635599475171, −8.479232159990344, −7.992311957706780, −7.200155163709679, −6.422791837207285, −5.727080294712974, −5.472237248785487, −4.492606060122733, −3.860746503867380, −2.909099653818042, −2.149379530114908, −1.068335379394225, 0, 1.068335379394225, 2.149379530114908, 2.909099653818042, 3.860746503867380, 4.492606060122733, 5.472237248785487, 5.727080294712974, 6.422791837207285, 7.200155163709679, 7.992311957706780, 8.479232159990344, 8.987635599475171, 9.894871311776432, 10.64193034879863, 10.86422319574373, 11.65109809558644, 11.93754075419907, 12.74925979699793, 13.49163247014587, 13.78049715599735, 14.37355364337805, 15.31092016975847, 15.62585477991060, 16.14178141825553, 16.82568913563879

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