Properties

Degree $2$
Conductor $1024$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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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

Degree: \(2\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $1$
Motivic weight: \(3\)
Character: $\chi_{1024} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1024,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.946763272\)
\(L(\frac12)\) \(\approx\) \(4.946763272\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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 \( 1 + 809.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.578360636327438504262297322471, −8.781597449875472427946833032009, −8.022329008446680396133092635051, −7.27909689024147577978778703305, −6.23283557968567083316537129701, −5.14982600369590871955751145236, −4.14116171119984349018329424448, −2.75412650814173250352747269890, −2.48697389390343085973778299815, −1.18487164950767454534058824419, 1.18487164950767454534058824419, 2.48697389390343085973778299815, 2.75412650814173250352747269890, 4.14116171119984349018329424448, 5.14982600369590871955751145236, 6.23283557968567083316537129701, 7.27909689024147577978778703305, 8.022329008446680396133092635051, 8.781597449875472427946833032009, 9.578360636327438504262297322471

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