Properties

Label 2-882-1.1-c3-0-17
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 − 3.89·5-s + 8·8-s − 7.79·10-s + 61.3·11-s − 53.6·13-s + 16·16-s + 32.1·17-s + 55.7·19-s − 15.5·20-s + 122.·22-s + 94.6·23-s − 109.·25-s − 107.·26-s − 138.·29-s − 132.·31-s + 32·32-s + 64.2·34-s + 149.·37-s + 111.·38-s − 31.1·40-s + 427.·41-s + 437.·43-s + 245.·44-s + 189.·46-s + 57.0·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.348·5-s + 0.353·8-s − 0.246·10-s + 1.68·11-s − 1.14·13-s + 0.250·16-s + 0.457·17-s + 0.673·19-s − 0.174·20-s + 1.18·22-s + 0.857·23-s − 0.878·25-s − 0.810·26-s − 0.884·29-s − 0.768·31-s + 0.176·32-s + 0.323·34-s + 0.662·37-s + 0.476·38-s − 0.123·40-s + 1.62·41-s + 1.55·43-s + 0.841·44-s + 0.606·46-s + 0.176·47-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

\(L(2)\) \(\approx\) \(3.397216935\)
\(L(\frac12)\) \(\approx\) \(3.397216935\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.89T + 125T^{2} \)
11 \( 1 - 61.3T + 1.33e3T^{2} \)
13 \( 1 + 53.6T + 2.19e3T^{2} \)
17 \( 1 - 32.1T + 4.91e3T^{2} \)
19 \( 1 - 55.7T + 6.85e3T^{2} \)
23 \( 1 - 94.6T + 1.21e4T^{2} \)
29 \( 1 + 138.T + 2.43e4T^{2} \)
31 \( 1 + 132.T + 2.97e4T^{2} \)
37 \( 1 - 149.T + 5.06e4T^{2} \)
41 \( 1 - 427.T + 6.89e4T^{2} \)
43 \( 1 - 437.T + 7.95e4T^{2} \)
47 \( 1 - 57.0T + 1.03e5T^{2} \)
53 \( 1 - 263.T + 1.48e5T^{2} \)
59 \( 1 - 451.T + 2.05e5T^{2} \)
61 \( 1 + 579.T + 2.26e5T^{2} \)
67 \( 1 - 309.T + 3.00e5T^{2} \)
71 \( 1 - 1.05e3T + 3.57e5T^{2} \)
73 \( 1 + 1.19e3T + 3.89e5T^{2} \)
79 \( 1 - 1.31e3T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3T + 5.71e5T^{2} \)
89 \( 1 + 233.T + 7.04e5T^{2} \)
97 \( 1 - 1.60e3T + 9.12e5T^{2} \)
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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

−9.556553922645206099061499193762, −9.178946293394297247613449360882, −7.69946421238976340555733443791, −7.23695780485060622098073355405, −6.19498659143027945997095611510, −5.32004243077272155792473801965, −4.24630351917497696373809426088, −3.55790985314990327580576680415, −2.29872978431818329262918064618, −0.944137019811539919812770333919, 0.944137019811539919812770333919, 2.29872978431818329262918064618, 3.55790985314990327580576680415, 4.24630351917497696373809426088, 5.32004243077272155792473801965, 6.19498659143027945997095611510, 7.23695780485060622098073355405, 7.69946421238976340555733443791, 9.178946293394297247613449360882, 9.556553922645206099061499193762

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