Properties

Label 2-882-1.1-c1-0-3
Degree $2$
Conductor $882$
Sign $1$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2.82·5-s − 8-s − 2.82·10-s + 2·11-s + 16-s − 1.41·17-s + 7.07·19-s + 2.82·20-s − 2·22-s + 4·23-s + 3.00·25-s − 2·29-s − 8.48·31-s − 32-s + 1.41·34-s + 10·37-s − 7.07·38-s − 2.82·40-s − 9.89·41-s + 2·43-s + 2·44-s − 4·46-s + 2.82·47-s − 3.00·50-s + 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.26·5-s − 0.353·8-s − 0.894·10-s + 0.603·11-s + 0.250·16-s − 0.342·17-s + 1.62·19-s + 0.632·20-s − 0.426·22-s + 0.834·23-s + 0.600·25-s − 0.371·29-s − 1.52·31-s − 0.176·32-s + 0.242·34-s + 1.64·37-s − 1.14·38-s − 0.447·40-s − 1.54·41-s + 0.304·43-s + 0.301·44-s − 0.589·46-s + 0.412·47-s − 0.424·50-s + 0.274·53-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: no
Arithmetic: yes
Character: $\chi_{882} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.510461815\)
\(L(\frac12)\) \(\approx\) \(1.510461815\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−9.769944579881218004385660802753, −9.495341183895255823792238557222, −8.714418903789153403984824790075, −7.55772417000874592739120469918, −6.78307803372587138832112294560, −5.87224703261557223156336786358, −5.09921554447328176462041806446, −3.52686130493695088434950481777, −2.29850546468802787455292500548, −1.20638853268741970222433572774, 1.20638853268741970222433572774, 2.29850546468802787455292500548, 3.52686130493695088434950481777, 5.09921554447328176462041806446, 5.87224703261557223156336786358, 6.78307803372587138832112294560, 7.55772417000874592739120469918, 8.714418903789153403984824790075, 9.495341183895255823792238557222, 9.769944579881218004385660802753

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