Properties

Label 2-882-7.4-c3-0-40
Degree $2$
Conductor $882$
Sign $0.968 + 0.250i$
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 + (6 + 10.3i)5-s − 7.99·8-s + (−12 + 20.7i)10-s + (24 − 41.5i)11-s − 56·13-s + (−8 − 13.8i)16-s + (57 − 98.7i)17-s + (1 + 1.73i)19-s − 48·20-s + 96·22-s + (−60 − 103. i)23-s + (−9.5 + 16.4i)25-s + (−56 − 96.9i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.536 + 0.929i)5-s − 0.353·8-s + (−0.379 + 0.657i)10-s + (0.657 − 1.13i)11-s − 1.19·13-s + (−0.125 − 0.216i)16-s + (0.813 − 1.40i)17-s + (0.0120 + 0.0209i)19-s − 0.536·20-s + 0.930·22-s + (−0.543 − 0.942i)23-s + (−0.0759 + 0.131i)25-s + (−0.422 − 0.731i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
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: \((2,\ 882,\ (\ :3/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.148812516\)
\(L(\frac12)\) \(\approx\) \(2.148812516\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6 - 10.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-24 + 41.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 56T + 2.19e3T^{2} \)
17 \( 1 + (-57 + 98.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (60 + 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 54T + 2.43e4T^{2} \)
31 \( 1 + (-118 + 204. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (73 + 126. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 126T + 6.89e4T^{2} \)
43 \( 1 + 376T + 7.95e4T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-87 + 150. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (69 - 119. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-190 - 329. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-242 + 419. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 576T + 3.57e5T^{2} \)
73 \( 1 + (575 - 995. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (388 + 672. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 378T + 5.71e5T^{2} \)
89 \( 1 + (-195 - 337. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−9.763792912353710620449209565304, −8.825543463473312514983365764538, −7.82263217262482659145075055596, −7.00780620700644231623354704790, −6.29173850874581406427328007487, −5.50607698969204585179748222972, −4.45486784181315420070080859124, −3.20636866839926964328854875056, −2.44698001843132376037635940762, −0.50340224632015936634852455496, 1.27736906756257935114135507374, 1.95990059510989396449734175574, 3.39515941283079181160923057581, 4.52646837043423810501434606529, 5.13158060715214338603530428372, 6.11346436661775963350640502755, 7.19254278864143682365348419750, 8.266613280009776784303157147613, 9.181115071686692938601913878219, 9.935701544149692363438345023187

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