Properties

Label 2-2e9-1.1-c1-0-9
Degree $2$
Conductor $512$
Sign $1$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 2.82·5-s + 4·7-s − 0.999·9-s + 1.41·11-s − 2.82·13-s + 4.00·15-s − 4·17-s − 7.07·19-s + 5.65·21-s + 4·23-s + 3.00·25-s − 5.65·27-s − 8.48·29-s + 8·31-s + 2.00·33-s + 11.3·35-s − 2.82·37-s − 4.00·39-s + 2·41-s + 4.24·43-s − 2.82·45-s + 9·49-s − 5.65·51-s − 2.82·53-s + 4.00·55-s − 10.0·57-s + ⋯
L(s)  = 1  + 0.816·3-s + 1.26·5-s + 1.51·7-s − 0.333·9-s + 0.426·11-s − 0.784·13-s + 1.03·15-s − 0.970·17-s − 1.62·19-s + 1.23·21-s + 0.834·23-s + 0.600·25-s − 1.08·27-s − 1.57·29-s + 1.43·31-s + 0.348·33-s + 1.91·35-s − 0.464·37-s − 0.640·39-s + 0.312·41-s + 0.646·43-s − 0.421·45-s + 1.28·49-s − 0.792·51-s − 0.388·53-s + 0.539·55-s − 1.32·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $1$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.355298260\)
\(L(\frac12)\) \(\approx\) \(2.355298260\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 7.07T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 2.82T + 53T^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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

−10.91209404595769029210519404696, −9.883314653812860460528287596748, −8.936781326194524102910309356880, −8.487612586958332116151505354964, −7.39445193201164554674056408186, −6.24521647187241645309584533271, −5.20880474696415961785815476276, −4.23899392513018733142803174210, −2.48954608600981803648358007327, −1.85171982909904310984154993292, 1.85171982909904310984154993292, 2.48954608600981803648358007327, 4.23899392513018733142803174210, 5.20880474696415961785815476276, 6.24521647187241645309584533271, 7.39445193201164554674056408186, 8.487612586958332116151505354964, 8.936781326194524102910309356880, 9.883314653812860460528287596748, 10.91209404595769029210519404696

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