Properties

Label 2-882-1.1-c3-0-2
Degree $2$
Conductor $882$
Sign $1$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
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 − 9.38·5-s − 8·8-s + 18.7·10-s − 20·11-s + 65.6·13-s + 16·16-s − 56.2·17-s + 9.38·19-s − 37.5·20-s + 40·22-s − 48·23-s − 37·25-s − 131.·26-s + 166·29-s − 206.·31-s − 32·32-s + 112.·34-s − 78·37-s − 18.7·38-s + 75.0·40-s − 393.·41-s + 436·43-s − 80·44-s + 96·46-s − 206.·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.839·5-s − 0.353·8-s + 0.593·10-s − 0.548·11-s + 1.40·13-s + 0.250·16-s − 0.803·17-s + 0.113·19-s − 0.419·20-s + 0.387·22-s − 0.435·23-s − 0.295·25-s − 0.990·26-s + 1.06·29-s − 1.19·31-s − 0.176·32-s + 0.567·34-s − 0.346·37-s − 0.0800·38-s + 0.296·40-s − 1.50·41-s + 1.54·43-s − 0.274·44-s + 0.307·46-s − 0.640·47-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: no
Arithmetic: yes
Character: $\chi_{882} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9328433019\)
\(L(\frac12)\) \(\approx\) \(0.9328433019\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 9.38T + 125T^{2} \)
11 \( 1 + 20T + 1.33e3T^{2} \)
13 \( 1 - 65.6T + 2.19e3T^{2} \)
17 \( 1 + 56.2T + 4.91e3T^{2} \)
19 \( 1 - 9.38T + 6.85e3T^{2} \)
23 \( 1 + 48T + 1.21e4T^{2} \)
29 \( 1 - 166T + 2.43e4T^{2} \)
31 \( 1 + 206.T + 2.97e4T^{2} \)
37 \( 1 + 78T + 5.06e4T^{2} \)
41 \( 1 + 393.T + 6.89e4T^{2} \)
43 \( 1 - 436T + 7.95e4T^{2} \)
47 \( 1 + 206.T + 1.03e5T^{2} \)
53 \( 1 + 62T + 1.48e5T^{2} \)
59 \( 1 - 666.T + 2.05e5T^{2} \)
61 \( 1 - 272.T + 2.26e5T^{2} \)
67 \( 1 - 580T + 3.00e5T^{2} \)
71 \( 1 - 544T + 3.57e5T^{2} \)
73 \( 1 + 600.T + 3.89e5T^{2} \)
79 \( 1 + 680T + 4.93e5T^{2} \)
83 \( 1 + 196.T + 5.71e5T^{2} \)
89 \( 1 - 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + 656.T + 9.12e5T^{2} \)
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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

−9.739796767729301659215511325273, −8.645554856818648397240544249760, −8.286108012076047630829941006302, −7.33497925738591060805815500058, −6.50336696420769706392207862451, −5.49013918813724257453180789283, −4.19094003270597575466085943541, −3.30616827862340583578236505598, −1.95778499536803532488249251807, −0.57944176600634645086332289842, 0.57944176600634645086332289842, 1.95778499536803532488249251807, 3.30616827862340583578236505598, 4.19094003270597575466085943541, 5.49013918813724257453180789283, 6.50336696420769706392207862451, 7.33497925738591060805815500058, 8.286108012076047630829941006302, 8.645554856818648397240544249760, 9.739796767729301659215511325273

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