Properties

Label 2-882-1.1-c3-0-6
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 − 19.7·5-s + 8·8-s − 39.5·10-s + 14·11-s − 50.9·13-s + 16·16-s + 1.41·17-s + 1.41·19-s − 79.1·20-s + 28·22-s − 140·23-s + 267·25-s − 101.·26-s + 286·29-s + 93.3·31-s + 32·32-s + 2.82·34-s − 38·37-s + 2.82·38-s − 158.·40-s − 125.·41-s − 34·43-s + 56·44-s − 280·46-s + 523.·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.77·5-s + 0.353·8-s − 1.25·10-s + 0.383·11-s − 1.08·13-s + 0.250·16-s + 0.0201·17-s + 0.0170·19-s − 0.885·20-s + 0.271·22-s − 1.26·23-s + 2.13·25-s − 0.768·26-s + 1.83·29-s + 0.540·31-s + 0.176·32-s + 0.0142·34-s − 0.168·37-s + 0.0120·38-s − 0.626·40-s − 0.479·41-s − 0.120·43-s + 0.191·44-s − 0.897·46-s + 1.62·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.919100433\)
\(L(\frac12)\) \(\approx\) \(1.919100433\)
\(L(\frac{5}{2})\) 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 \( 1 + 19.7T + 125T^{2} \)
11 \( 1 - 14T + 1.33e3T^{2} \)
13 \( 1 + 50.9T + 2.19e3T^{2} \)
17 \( 1 - 1.41T + 4.91e3T^{2} \)
19 \( 1 - 1.41T + 6.85e3T^{2} \)
23 \( 1 + 140T + 1.21e4T^{2} \)
29 \( 1 - 286T + 2.43e4T^{2} \)
31 \( 1 - 93.3T + 2.97e4T^{2} \)
37 \( 1 + 38T + 5.06e4T^{2} \)
41 \( 1 + 125.T + 6.89e4T^{2} \)
43 \( 1 + 34T + 7.95e4T^{2} \)
47 \( 1 - 523.T + 1.03e5T^{2} \)
53 \( 1 - 74T + 1.48e5T^{2} \)
59 \( 1 - 434.T + 2.05e5T^{2} \)
61 \( 1 + 14.1T + 2.26e5T^{2} \)
67 \( 1 - 684T + 3.00e5T^{2} \)
71 \( 1 + 588T + 3.57e5T^{2} \)
73 \( 1 - 270.T + 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 422.T + 5.71e5T^{2} \)
89 \( 1 - 618.T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−9.946544502531796978791203499092, −8.640081481245795414438356933812, −7.919657262481967479203518122031, −7.20133251311461425350983208211, −6.39083494036806901929692551802, −5.03989916796932359090620313456, −4.31074968961621518905078370274, −3.56379522214150126113651148778, −2.46627163507187380892450697437, −0.66752804351861086521769475915, 0.66752804351861086521769475915, 2.46627163507187380892450697437, 3.56379522214150126113651148778, 4.31074968961621518905078370274, 5.03989916796932359090620313456, 6.39083494036806901929692551802, 7.20133251311461425350983208211, 7.919657262481967479203518122031, 8.640081481245795414438356933812, 9.946544502531796978791203499092

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