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  + 2.80·3-s + 0.844·5-s − 29.0·7-s − 19.1·9-s + 17.1·11-s + 68.6·13-s + 2.36·15-s − 86.7·17-s + 77.5·19-s − 81.5·21-s + 70.2·23-s − 124.·25-s − 129.·27-s + 89.6·29-s − 8.86·31-s + 48.1·33-s − 24.5·35-s + 30.7·37-s + 192.·39-s − 153.·41-s − 171.·43-s − 16.1·45-s + 99.9·47-s + 502.·49-s − 243.·51-s + 550.·53-s + 14.4·55-s + ⋯
L(s)  = 1  + 0.539·3-s + 0.0754·5-s − 1.57·7-s − 0.708·9-s + 0.470·11-s + 1.46·13-s + 0.0407·15-s − 1.23·17-s + 0.936·19-s − 0.847·21-s + 0.636·23-s − 0.994·25-s − 0.922·27-s + 0.574·29-s − 0.0513·31-s + 0.253·33-s − 0.118·35-s + 0.136·37-s + 0.791·39-s − 0.583·41-s − 0.606·43-s − 0.0534·45-s + 0.310·47-s + 1.46·49-s − 0.667·51-s + 1.42·53-s + 0.0354·55-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  =  \((2,\ 1024,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.881359101\)
\(L(\frac12)\)  \(\approx\)  \(1.881359101\)
\(L(\frac{5}{2})\)   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 \( 1 - 2.80T + 27T^{2} \)
5 \( 1 - 0.844T + 125T^{2} \)
7 \( 1 + 29.0T + 343T^{2} \)
11 \( 1 - 17.1T + 1.33e3T^{2} \)
13 \( 1 - 68.6T + 2.19e3T^{2} \)
17 \( 1 + 86.7T + 4.91e3T^{2} \)
19 \( 1 - 77.5T + 6.85e3T^{2} \)
23 \( 1 - 70.2T + 1.21e4T^{2} \)
29 \( 1 - 89.6T + 2.43e4T^{2} \)
31 \( 1 + 8.86T + 2.97e4T^{2} \)
37 \( 1 - 30.7T + 5.06e4T^{2} \)
41 \( 1 + 153.T + 6.89e4T^{2} \)
43 \( 1 + 171.T + 7.95e4T^{2} \)
47 \( 1 - 99.9T + 1.03e5T^{2} \)
53 \( 1 - 550.T + 1.48e5T^{2} \)
59 \( 1 - 459.T + 2.05e5T^{2} \)
61 \( 1 + 0.479T + 2.26e5T^{2} \)
67 \( 1 - 799.T + 3.00e5T^{2} \)
71 \( 1 - 419.T + 3.57e5T^{2} \)
73 \( 1 - 374.T + 3.89e5T^{2} \)
79 \( 1 - 705.T + 4.93e5T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 - 4.72T + 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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

−9.341638080152952844898124209649, −8.913989423838868746274899886899, −8.087783512279566455454404537814, −6.81611226366816001918102854781, −6.33174488729863046080835777908, −5.40686256970324634991315196722, −3.84862742969343717921318423960, −3.34533140045411204362972168859, −2.27974282296017960222069509311, −0.69109870301647899797521411219, 0.69109870301647899797521411219, 2.27974282296017960222069509311, 3.34533140045411204362972168859, 3.84862742969343717921318423960, 5.40686256970324634991315196722, 6.33174488729863046080835777908, 6.81611226366816001918102854781, 8.087783512279566455454404537814, 8.913989423838868746274899886899, 9.341638080152952844898124209649

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