↑

Properties

Label 2-882-7.2-c3-0-16
Degree $2$
Conductor $882$
Sign $0.0725 - 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (7.94 − 13.7i)5-s + 7.99·8-s + (15.8 + 27.5i)10-s + (28.6 + 49.7i)11-s − 5.69·13-s + (−8 + 13.8i)16-s + (25.9 + 44.9i)17-s + (8.10 − 14.0i)19-s − 63.5·20-s − 114.·22-s + (−106. + 184. i)23-s + (−63.8 − 110. i)25-s + (5.69 − 9.87i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.711 − 1.23i)5-s + 0.353·8-s + (0.502 + 0.870i)10-s + (0.786 + 1.36i)11-s − 0.121·13-s + (−0.125 + 0.216i)16-s + (0.370 + 0.641i)17-s + (0.0978 − 0.169i)19-s − 0.711·20-s − 1.11·22-s + (−0.967 + 1.67i)23-s + (−0.511 − 0.885i)25-s + (0.0429 − 0.0744i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0725 - 0.997i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.668128497\)
\(L(\frac12)\) \(\approx\) \(1.668128497\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.94 + 13.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.6 - 49.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 5.69T + 2.19e3T^{2} \)
17 \( 1 + (-25.9 - 44.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-8.10 + 14.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (106. - 184. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 218.T + 2.43e4T^{2} \)
31 \( 1 + (125. + 217. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (193. - 334. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 328.T + 6.89e4T^{2} \)
43 \( 1 + 37.5T + 7.95e4T^{2} \)
47 \( 1 + (-127. + 220. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-105. - 183. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-206. - 356. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (418. - 724. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-82.7 - 143. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 465.T + 3.57e5T^{2} \)
73 \( 1 + (-224. - 389. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-171. + 297. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.50e3T + 5.71e5T^{2} \)
89 \( 1 + (-170. + 295. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 865.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.766990062522566554475051208958, −9.142251340377031593429171814693, −8.338829263381755825251085288302, −7.45500204976840681712335800772, −6.50647493697581420170059855958, −5.58153866508287147135356162744, −4.84968757111643896808808674447, −3.89830855470759996823196146861, −1.93029491489534888888708437319, −1.18600860180194120205756681436, 0.52383656363699570609374607014, 1.95733691902567105454720029681, 2.97212136590908617342758672746, 3.68955955974685610651532756226, 5.15270635466257768416253548168, 6.33518180456011851981945628808, 6.78520128122529180076469889957, 8.046198638663704270363604045397, 8.845851164065563733798927268326, 9.686419320199500737375235570582

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