Properties

Degree 2
Conductor $ 2^{10} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 8.43·3-s − 12.2·5-s + 1.63·7-s + 44.1·9-s + 25.7·11-s + 13.2·13-s + 103.·15-s − 53.6·17-s − 100.·19-s − 13.8·21-s + 25.1·23-s + 25.6·25-s − 144.·27-s − 256.·29-s + 132.·31-s − 217.·33-s − 20.1·35-s − 247.·37-s − 111.·39-s + 198.·41-s − 404.·43-s − 542.·45-s − 78.3·47-s − 340.·49-s + 452.·51-s − 743.·53-s − 315.·55-s + ⋯
L(s)  = 1  − 1.62·3-s − 1.09·5-s + 0.0885·7-s + 1.63·9-s + 0.705·11-s + 0.282·13-s + 1.78·15-s − 0.764·17-s − 1.21·19-s − 0.143·21-s + 0.227·23-s + 0.205·25-s − 1.03·27-s − 1.63·29-s + 0.768·31-s − 1.14·33-s − 0.0971·35-s − 1.09·37-s − 0.457·39-s + 0.756·41-s − 1.43·43-s − 1.79·45-s − 0.243·47-s − 0.992·49-s + 1.24·51-s − 1.92·53-s − 0.774·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1024\)    =    \(2^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{1024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1024,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(0.3825353315\)
\(L(\frac12)\)  \(\approx\)  \(0.3825353315\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 8.43T + 27T^{2} \)
5 \( 1 + 12.2T + 125T^{2} \)
7 \( 1 - 1.63T + 343T^{2} \)
11 \( 1 - 25.7T + 1.33e3T^{2} \)
13 \( 1 - 13.2T + 2.19e3T^{2} \)
17 \( 1 + 53.6T + 4.91e3T^{2} \)
19 \( 1 + 100.T + 6.85e3T^{2} \)
23 \( 1 - 25.1T + 1.21e4T^{2} \)
29 \( 1 + 256.T + 2.43e4T^{2} \)
31 \( 1 - 132.T + 2.97e4T^{2} \)
37 \( 1 + 247.T + 5.06e4T^{2} \)
41 \( 1 - 198.T + 6.89e4T^{2} \)
43 \( 1 + 404.T + 7.95e4T^{2} \)
47 \( 1 + 78.3T + 1.03e5T^{2} \)
53 \( 1 + 743.T + 1.48e5T^{2} \)
59 \( 1 + 65.8T + 2.05e5T^{2} \)
61 \( 1 - 273.T + 2.26e5T^{2} \)
67 \( 1 - 399.T + 3.00e5T^{2} \)
71 \( 1 - 727.T + 3.57e5T^{2} \)
73 \( 1 - 106.T + 3.89e5T^{2} \)
79 \( 1 - 58.9T + 4.93e5T^{2} \)
83 \( 1 - 580.T + 5.71e5T^{2} \)
89 \( 1 + 768.T + 7.04e5T^{2} \)
97 \( 1 + 809.T + 9.12e5T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.723098431634566575855113286121, −8.674061083271970067722166475784, −7.78394369683985661769811955613, −6.72601131264726556545849026940, −6.33909660573111759265656738157, −5.18070911936530988800275319731, −4.39749519119970754136259074923, −3.62222867414856132771003140515, −1.71493998677633296920465832346, −0.35776384759393791392904697280, 0.35776384759393791392904697280, 1.71493998677633296920465832346, 3.62222867414856132771003140515, 4.39749519119970754136259074923, 5.18070911936530988800275319731, 6.33909660573111759265656738157, 6.72601131264726556545849026940, 7.78394369683985661769811955613, 8.674061083271970067722166475784, 9.723098431634566575855113286121

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