↑

Properties

Label 2-882-7.2-c3-0-45
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (4.5 − 7.79i)5-s − 7.99·8-s + (−9 − 15.5i)10-s + (−28.5 − 49.3i)11-s + 70·13-s + (−8 + 13.8i)16-s + (−25.5 − 44.1i)17-s + (2.5 − 4.33i)19-s − 36·20-s − 114·22-s + (34.5 − 59.7i)23-s + (22 + 38.1i)25-s + (70 − 121. i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.402 − 0.697i)5-s − 0.353·8-s + (−0.284 − 0.492i)10-s + (−0.781 − 1.35i)11-s + 1.49·13-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + (0.0301 − 0.0522i)19-s − 0.402·20-s − 1.10·22-s + (0.312 − 0.541i)23-s + (0.175 + 0.304i)25-s + (0.528 − 0.914i)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: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.869677161\)
\(L(\frac12)\) \(\approx\) \(1.869677161\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.5 + 7.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (28.5 + 49.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 70T + 2.19e3T^{2} \)
17 \( 1 + (25.5 + 44.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-34.5 + 59.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 114T + 2.43e4T^{2} \)
31 \( 1 + (-11.5 - 19.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-126.5 + 219. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (100.5 - 174. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (196.5 + 340. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (109.5 + 189. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (354.5 - 614. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (209.5 + 362. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 96T + 3.57e5T^{2} \)
73 \( 1 + (156.5 + 271. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (230.5 - 399. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 588T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 + 880. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.83e3T + 9.12e5T^{2} \)
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

−9.147409219671262156311145214390, −8.785101259799875944939989603785, −7.81354504870175813919754011897, −6.39115237548035909812671913911, −5.66057308737277947206967451655, −4.88932624879956249380394581299, −3.71313759502123318246440156023, −2.80326950899123060341505491075, −1.44475935147327942645373198978, −0.42438150281191247574806351189, 1.67020384035870446609960761170, 2.89158597003839989872205287195, 3.98379792382082876720006959059, 4.98074150014480016222205894316, 6.01412956470449521866673363870, 6.64156583006287359659093383740, 7.53449368960982168236250686623, 8.326203716411099886365180275606, 9.315033098008783370951175670099, 10.22248605004855437213955097624

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