Properties

Label 2-882-1.1-c3-0-20
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 + 15.8·5-s + 8·8-s + 31.7·10-s − 57.3·11-s + 5.69·13-s + 16·16-s + 51.8·17-s + 16.2·19-s + 63.5·20-s − 114.·22-s + 213.·23-s + 127.·25-s + 11.3·26-s + 218.·29-s − 251.·31-s + 32·32-s + 103.·34-s + 386.·37-s + 32.4·38-s + 127.·40-s + 328.·41-s − 37.5·43-s − 229.·44-s + 426.·46-s + 254.·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.42·5-s + 0.353·8-s + 1.00·10-s − 1.57·11-s + 0.121·13-s + 0.250·16-s + 0.740·17-s + 0.195·19-s + 0.711·20-s − 1.11·22-s + 1.93·23-s + 1.02·25-s + 0.0859·26-s + 1.39·29-s − 1.45·31-s + 0.176·32-s + 0.523·34-s + 1.71·37-s + 0.138·38-s + 0.502·40-s + 1.25·41-s − 0.133·43-s − 0.786·44-s + 1.36·46-s + 0.791·47-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

\(L(2)\) \(\approx\) \(4.397168466\)
\(L(\frac12)\) \(\approx\) \(4.397168466\)
\(L(\frac{5}{2})\) 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 \( 1 - 15.8T + 125T^{2} \)
11 \( 1 + 57.3T + 1.33e3T^{2} \)
13 \( 1 - 5.69T + 2.19e3T^{2} \)
17 \( 1 - 51.8T + 4.91e3T^{2} \)
19 \( 1 - 16.2T + 6.85e3T^{2} \)
23 \( 1 - 213.T + 1.21e4T^{2} \)
29 \( 1 - 218.T + 2.43e4T^{2} \)
31 \( 1 + 251.T + 2.97e4T^{2} \)
37 \( 1 - 386.T + 5.06e4T^{2} \)
41 \( 1 - 328.T + 6.89e4T^{2} \)
43 \( 1 + 37.5T + 7.95e4T^{2} \)
47 \( 1 - 254.T + 1.03e5T^{2} \)
53 \( 1 + 211.T + 1.48e5T^{2} \)
59 \( 1 - 412.T + 2.05e5T^{2} \)
61 \( 1 + 836.T + 2.26e5T^{2} \)
67 \( 1 + 165.T + 3.00e5T^{2} \)
71 \( 1 - 465.T + 3.57e5T^{2} \)
73 \( 1 - 449.T + 3.89e5T^{2} \)
79 \( 1 + 343.T + 4.93e5T^{2} \)
83 \( 1 + 1.50e3T + 5.71e5T^{2} \)
89 \( 1 - 341.T + 7.04e5T^{2} \)
97 \( 1 + 865.T + 9.12e5T^{2} \)
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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

−9.864783963039341036748943054560, −9.073222236486308508142557890791, −7.899962458693330902949627752818, −7.06001676019769261432222178661, −5.96101408151034563092602807112, −5.43890016548073156820113994508, −4.65211902230538311190657271746, −3.05861659661790534892130539246, −2.42743037877594988650576148684, −1.09765747474199421275310300103, 1.09765747474199421275310300103, 2.42743037877594988650576148684, 3.05861659661790534892130539246, 4.65211902230538311190657271746, 5.43890016548073156820113994508, 5.96101408151034563092602807112, 7.06001676019769261432222178661, 7.899962458693330902949627752818, 9.073222236486308508142557890791, 9.864783963039341036748943054560

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