Properties

Degree $2$
Conductor $882$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 3·11-s + 2·13-s + 16-s − 6·17-s + 2·19-s − 3·20-s − 3·22-s + 6·23-s + 4·25-s − 2·26-s − 9·29-s − 7·31-s − 32-s + 6·34-s − 10·37-s − 2·38-s + 3·40-s − 4·43-s + 3·44-s − 6·46-s − 12·47-s − 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 0.904·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.670·20-s − 0.639·22-s + 1.25·23-s + 4/5·25-s − 0.392·26-s − 1.67·29-s − 1.25·31-s − 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.324·38-s + 0.474·40-s − 0.609·43-s + 0.452·44-s − 0.884·46-s − 1.75·47-s − 0.565·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{882} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−9.438664945874852130105936905544, −8.882087049269418887856086132720, −8.143428262544059580201771154193, −7.14400221617286508047479839089, −6.70720744145358122288448799487, −5.30135083620848369798403743062, −4.05162352634345833644814800792, −3.32944386865340105186531370454, −1.64799092613985375020716144538, 0, 1.64799092613985375020716144538, 3.32944386865340105186531370454, 4.05162352634345833644814800792, 5.30135083620848369798403743062, 6.70720744145358122288448799487, 7.14400221617286508047479839089, 8.143428262544059580201771154193, 8.882087049269418887856086132720, 9.438664945874852130105936905544

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