Properties

Label 2-882-1.1-c1-0-12
Degree $2$
Conductor $882$
Sign $-1$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
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  − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 4·11-s − 6·13-s + 16-s + 2·17-s + 4·19-s − 2·20-s − 4·22-s − 8·23-s − 25-s + 6·26-s + 2·29-s − 32-s − 2·34-s − 10·37-s − 4·38-s + 2·40-s − 6·41-s − 4·43-s + 4·44-s + 8·46-s + 50-s − 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s − 0.852·22-s − 1.66·23-s − 1/5·25-s + 1.17·26-s + 0.371·29-s − 0.176·32-s − 0.342·34-s − 1.64·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s − 0.609·43-s + 0.603·44-s + 1.17·46-s + 0.141·50-s − 0.832·52-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: $-1$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.804201361737724343194425540841, −8.882543700432626086795887126363, −7.954464142207805330736993023418, −7.36399840290663473896741338506, −6.54855688605176886120203881032, −5.32455191890654195761811472253, −4.16943092436514571048096348777, −3.17941538941685381741709304651, −1.71238689353652455033144003099, 0, 1.71238689353652455033144003099, 3.17941538941685381741709304651, 4.16943092436514571048096348777, 5.32455191890654195761811472253, 6.54855688605176886120203881032, 7.36399840290663473896741338506, 7.954464142207805330736993023418, 8.882543700432626086795887126363, 9.804201361737724343194425540841

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