Properties

Label 2-882-1.1-c3-0-22
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 6·5-s + 8·8-s + 12·10-s + 30·11-s + 53·13-s + 16·16-s + 84·17-s − 97·19-s + 24·20-s + 60·22-s − 84·23-s − 89·25-s + 106·26-s + 180·29-s + 179·31-s + 32·32-s + 168·34-s − 145·37-s − 194·38-s + 48·40-s − 126·41-s − 325·43-s + 120·44-s − 168·46-s + 366·47-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 + 1.13·13-s + 1/4·16-s + 1.19·17-s − 1.17·19-s + 0.268·20-s + 0.581·22-s − 0.761·23-s − 0.711·25-s + 0.799·26-s + 1.15·29-s + 1.03·31-s + 0.176·32-s + 0.847·34-s − 0.644·37-s − 0.828·38-s + 0.189·40-s − 0.479·41-s − 1.15·43-s + 0.411·44-s − 0.538·46-s + 1.13·47-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: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.219559458\)
\(L(\frac12)\) \(\approx\) \(4.219559458\)
\(L(\frac{5}{2})\) 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 \( 1 - 53 T + p^{3} T^{2} \)
17 \( 1 - 84 T + p^{3} T^{2} \)
19 \( 1 + 97 T + p^{3} T^{2} \)
23 \( 1 + 84 T + p^{3} T^{2} \)
29 \( 1 - 180 T + p^{3} T^{2} \)
31 \( 1 - 179 T + p^{3} T^{2} \)
37 \( 1 + 145 T + p^{3} T^{2} \)
41 \( 1 + 126 T + p^{3} T^{2} \)
43 \( 1 + 325 T + p^{3} T^{2} \)
47 \( 1 - 366 T + p^{3} T^{2} \)
53 \( 1 - 768 T + p^{3} T^{2} \)
59 \( 1 - 264 T + p^{3} T^{2} \)
61 \( 1 - 818 T + p^{3} T^{2} \)
67 \( 1 + 523 T + p^{3} T^{2} \)
71 \( 1 - 342 T + p^{3} T^{2} \)
73 \( 1 + 43 T + p^{3} T^{2} \)
79 \( 1 + 1171 T + p^{3} T^{2} \)
83 \( 1 - 810 T + p^{3} T^{2} \)
89 \( 1 - 600 T + p^{3} T^{2} \)
97 \( 1 - 386 T + p^{3} T^{2} \)
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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

−10.05506942637359343655414795719, −8.787148522613480694708585065152, −8.142565341057299212836313481607, −6.85660579523683537053121006690, −6.18924324874339653487200823264, −5.48521086229103939987071304482, −4.24976690707695419636168562726, −3.51430818787618737544820718574, −2.20303142741957992740104388077, −1.09792233414450520517346022926, 1.09792233414450520517346022926, 2.20303142741957992740104388077, 3.51430818787618737544820718574, 4.24976690707695419636168562726, 5.48521086229103939987071304482, 6.18924324874339653487200823264, 6.85660579523683537053121006690, 8.142565341057299212836313481607, 8.787148522613480694708585065152, 10.05506942637359343655414795719

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