Properties

Label 2-882-1.1-c3-0-25
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 15.2·5-s + 8·8-s + 30.4·10-s + 2·11-s − 30.4·13-s + 16·16-s − 45.6·17-s + 152.·19-s + 60.9·20-s + 4·22-s + 30·23-s + 107.·25-s − 60.9·26-s + 212·29-s + 213.·31-s + 32·32-s − 91.3·34-s + 246·37-s + 304.·38-s + 121.·40-s − 319.·41-s − 284·43-s + 8·44-s + 60·46-s + 60.9·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.36·5-s + 0.353·8-s + 0.963·10-s + 0.0548·11-s − 0.649·13-s + 0.250·16-s − 0.651·17-s + 1.83·19-s + 0.681·20-s + 0.0387·22-s + 0.271·23-s + 0.856·25-s − 0.459·26-s + 1.35·29-s + 1.23·31-s + 0.176·32-s − 0.460·34-s + 1.09·37-s + 1.30·38-s + 0.481·40-s − 1.21·41-s − 1.00·43-s + 0.0274·44-s + 0.192·46-s + 0.189·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.541105677\)
\(L(\frac12)\) \(\approx\) \(4.541105677\)
\(L(\frac{5}{2})\) 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 \( 1 - 15.2T + 125T^{2} \)
11 \( 1 - 2T + 1.33e3T^{2} \)
13 \( 1 + 30.4T + 2.19e3T^{2} \)
17 \( 1 + 45.6T + 4.91e3T^{2} \)
19 \( 1 - 152.T + 6.85e3T^{2} \)
23 \( 1 - 30T + 1.21e4T^{2} \)
29 \( 1 - 212T + 2.43e4T^{2} \)
31 \( 1 - 213.T + 2.97e4T^{2} \)
37 \( 1 - 246T + 5.06e4T^{2} \)
41 \( 1 + 319.T + 6.89e4T^{2} \)
43 \( 1 + 284T + 7.95e4T^{2} \)
47 \( 1 - 60.9T + 1.03e5T^{2} \)
53 \( 1 - 548T + 1.48e5T^{2} \)
59 \( 1 + 670.T + 2.05e5T^{2} \)
61 \( 1 + 517.T + 2.26e5T^{2} \)
67 \( 1 - 652T + 3.00e5T^{2} \)
71 \( 1 - 770T + 3.57e5T^{2} \)
73 \( 1 + 974.T + 3.89e5T^{2} \)
79 \( 1 - 472T + 4.93e5T^{2} \)
83 \( 1 - 182.T + 5.71e5T^{2} \)
89 \( 1 - 715.T + 7.04e5T^{2} \)
97 \( 1 - 304.T + 9.12e5T^{2} \)
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

−9.875103238905782947370440358810, −9.113517015509372414785158950541, −7.949124826792866054137027542024, −6.88696143129432099711233691745, −6.21614586101923405045818994879, −5.27607605348031761811521296104, −4.65264119280266283289923430255, −3.15625524286318340802049901288, −2.33012378638829335539621981084, −1.13332601031539224664623395241, 1.13332601031539224664623395241, 2.33012378638829335539621981084, 3.15625524286318340802049901288, 4.65264119280266283289923430255, 5.27607605348031761811521296104, 6.21614586101923405045818994879, 6.88696143129432099711233691745, 7.949124826792866054137027542024, 9.113517015509372414785158950541, 9.875103238905782947370440358810

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