Properties

Label 2-882-1.1-c1-0-8
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 $0$

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: $1$
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: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.017796288\)
\(L(\frac12)\) \(\approx\) \(3.017796288\)
\(L(\frac{3}{2})\) 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

−10.26872983008473533451567565203, −9.521561835609810671858571335000, −8.388380094054281186601061743616, −7.49053770083860455326674259398, −6.40196296984473786838190696766, −5.62582070914781549215591544481, −5.13267260304391847265082866628, −3.69829470531395738272962092099, −2.65191870625457881403725845166, −1.52906039199785314571915081352, 1.52906039199785314571915081352, 2.65191870625457881403725845166, 3.69829470531395738272962092099, 5.13267260304391847265082866628, 5.62582070914781549215591544481, 6.40196296984473786838190696766, 7.49053770083860455326674259398, 8.388380094054281186601061743616, 9.521561835609810671858571335000, 10.26872983008473533451567565203

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