Properties

Label 2-882-1.1-c3-0-34
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 6·5-s − 8·8-s − 12·10-s − 30·11-s − 2·13-s + 16·16-s + 66·17-s + 52·19-s + 24·20-s + 60·22-s − 114·23-s − 89·25-s + 4·26-s − 72·29-s + 196·31-s − 32·32-s − 132·34-s − 286·37-s − 104·38-s − 48·40-s − 378·41-s + 164·43-s − 120·44-s + 228·46-s − 228·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.536·5-s − 0.353·8-s − 0.379·10-s − 0.822·11-s − 0.0426·13-s + 1/4·16-s + 0.941·17-s + 0.627·19-s + 0.268·20-s + 0.581·22-s − 1.03·23-s − 0.711·25-s + 0.0301·26-s − 0.461·29-s + 1.13·31-s − 0.176·32-s − 0.665·34-s − 1.27·37-s − 0.443·38-s − 0.189·40-s − 1.43·41-s + 0.581·43-s − 0.411·44-s + 0.730·46-s − 0.707·47-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 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 + 30 T + p^{3} T^{2} \)
13 \( 1 + 2 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 - 52 T + p^{3} T^{2} \)
23 \( 1 + 114 T + p^{3} T^{2} \)
29 \( 1 + 72 T + p^{3} T^{2} \)
31 \( 1 - 196 T + p^{3} T^{2} \)
37 \( 1 + 286 T + p^{3} T^{2} \)
41 \( 1 + 378 T + p^{3} T^{2} \)
43 \( 1 - 164 T + p^{3} T^{2} \)
47 \( 1 + 228 T + p^{3} T^{2} \)
53 \( 1 - 348 T + p^{3} T^{2} \)
59 \( 1 + 348 T + p^{3} T^{2} \)
61 \( 1 - 106 T + p^{3} T^{2} \)
67 \( 1 - 596 T + p^{3} T^{2} \)
71 \( 1 + 630 T + p^{3} T^{2} \)
73 \( 1 - 1042 T + p^{3} T^{2} \)
79 \( 1 + 88 T + p^{3} T^{2} \)
83 \( 1 + 1440 T + p^{3} T^{2} \)
89 \( 1 - 1374 T + p^{3} T^{2} \)
97 \( 1 - 34 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.558998757719835334369672290920, −8.383214222939481173119503532025, −7.82639343617231311496367007621, −6.87414883245019214785990900827, −5.84607362542699487640131407267, −5.15000225408950394502137665612, −3.64332908176095657160530089066, −2.51601187284871032215428564529, −1.43666607632335274257302355314, 0, 1.43666607632335274257302355314, 2.51601187284871032215428564529, 3.64332908176095657160530089066, 5.15000225408950394502137665612, 5.84607362542699487640131407267, 6.87414883245019214785990900827, 7.82639343617231311496367007621, 8.383214222939481173119503532025, 9.558998757719835334369672290920

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