Properties

Label 2-882-1.1-c3-0-38
Degree $2$
Conductor $882$
Sign $-1$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s − 15·5-s + 8·8-s − 30·10-s + 9·11-s + 88·13-s + 16·16-s − 84·17-s − 104·19-s − 60·20-s + 18·22-s + 84·23-s + 100·25-s + 176·26-s − 51·29-s − 185·31-s + 32·32-s − 168·34-s + 44·37-s − 208·38-s − 120·40-s − 168·41-s + 326·43-s + 36·44-s + 168·46-s − 138·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.246·11-s + 1.87·13-s + 1/4·16-s − 1.19·17-s − 1.25·19-s − 0.670·20-s + 0.174·22-s + 0.761·23-s + 4/5·25-s + 1.32·26-s − 0.326·29-s − 1.07·31-s + 0.176·32-s − 0.847·34-s + 0.195·37-s − 0.887·38-s − 0.474·40-s − 0.639·41-s + 1.15·43-s + 0.123·44-s + 0.538·46-s − 0.428·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3 p T + p^{3} T^{2} \)
11 \( 1 - 9 T + p^{3} T^{2} \)
13 \( 1 - 88 T + p^{3} T^{2} \)
17 \( 1 + 84 T + p^{3} T^{2} \)
19 \( 1 + 104 T + p^{3} T^{2} \)
23 \( 1 - 84 T + p^{3} T^{2} \)
29 \( 1 + 51 T + p^{3} T^{2} \)
31 \( 1 + 185 T + p^{3} T^{2} \)
37 \( 1 - 44 T + p^{3} T^{2} \)
41 \( 1 + 168 T + p^{3} T^{2} \)
43 \( 1 - 326 T + p^{3} T^{2} \)
47 \( 1 + 138 T + p^{3} T^{2} \)
53 \( 1 + 639 T + p^{3} T^{2} \)
59 \( 1 - 159 T + p^{3} T^{2} \)
61 \( 1 + 722 T + p^{3} T^{2} \)
67 \( 1 + 166 T + p^{3} T^{2} \)
71 \( 1 + 1086 T + p^{3} T^{2} \)
73 \( 1 + 218 T + p^{3} T^{2} \)
79 \( 1 + 583 T + p^{3} T^{2} \)
83 \( 1 + 597 T + p^{3} T^{2} \)
89 \( 1 + 1038 T + p^{3} T^{2} \)
97 \( 1 - 169 T + p^{3} T^{2} \)
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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

−9.020372877160189866594374323930, −8.500143141755566242150400988715, −7.53748870504286197421930659037, −6.65046736021345723278512106236, −5.90553592157913114726270144286, −4.51123877423003142686403288154, −3.99468304823770050978431227600, −3.11085711326258579644479565788, −1.57428354268467139153562611886, 0, 1.57428354268467139153562611886, 3.11085711326258579644479565788, 3.99468304823770050978431227600, 4.51123877423003142686403288154, 5.90553592157913114726270144286, 6.65046736021345723278512106236, 7.53748870504286197421930659037, 8.500143141755566242150400988715, 9.020372877160189866594374323930

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