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Properties

Label 2-882-7.4-c3-0-49
Degree $2$
Conductor $882$
Sign $-0.991 + 0.126i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−10.4 − 18.0i)5-s − 7.99·8-s + (20.8 − 36.0i)10-s + (7.58 − 13.1i)11-s − 2.16·13-s + (−8 − 13.8i)16-s + (59.6 − 103. i)17-s + (−16.7 − 29.0i)19-s + 83.3·20-s + 30.3·22-s + (0.325 + 0.564i)23-s + (−154. + 267. i)25-s + (−2.16 − 3.74i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.931 − 1.61i)5-s − 0.353·8-s + (0.658 − 1.14i)10-s + (0.207 − 0.359i)11-s − 0.0461·13-s + (−0.125 − 0.216i)16-s + (0.851 − 1.47i)17-s + (−0.202 − 0.350i)19-s + 0.931·20-s + 0.293·22-s + (0.00295 + 0.00511i)23-s + (−1.23 + 2.14i)25-s + (−0.0163 − 0.0282i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4244967235\)
\(L(\frac12)\) \(\approx\) \(0.4244967235\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (10.4 + 18.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-7.58 + 13.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 2.16T + 2.19e3T^{2} \)
17 \( 1 + (-59.6 + 103. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (16.7 + 29.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-0.325 - 0.564i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 163.T + 2.43e4T^{2} \)
31 \( 1 + (111. - 193. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (84.2 + 145. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 323.T + 6.89e4T^{2} \)
43 \( 1 - 221.T + 7.95e4T^{2} \)
47 \( 1 + (254. + 439. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (88.2 - 152. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (227. - 393. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-19.3 - 33.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (70.8 - 122. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 602.T + 3.57e5T^{2} \)
73 \( 1 + (551. - 954. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-58.1 - 100. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 568.T + 5.71e5T^{2} \)
89 \( 1 + (-191. - 331. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 334.T + 9.12e5T^{2} \)
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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

−8.917386529620331214954540869442, −8.591166711443351474395347965593, −7.62798231974978748224691660041, −6.93146339375968261213104374642, −5.52414333908519000875737740791, −4.98756530323137534015746258608, −4.14462611105665257581236173586, −3.11936717356964357497432543143, −1.16898988492149796530960660491, −0.10699929065387199215873938850, 1.71194270310842987835749204537, 2.95933247193422677382675170296, 3.64453363529124792443397353958, 4.48282589900028257719801066058, 5.96050089426310260449776166766, 6.62155424679114290935290741838, 7.64447981795478706168919396089, 8.296618571590731521671677366318, 9.678038706885603602018055906733, 10.42452459092125436118524370843

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