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  + 4.62·3-s − 17.8·5-s + 13.8·7-s − 5.59·9-s − 2.18·11-s + 46.3·13-s − 82.7·15-s − 18.6·17-s − 122.·19-s + 64.1·21-s + 134.·23-s + 194.·25-s − 150.·27-s + 84.5·29-s − 31.5·31-s − 10.1·33-s − 248.·35-s + 126.·37-s + 214.·39-s + 210.·41-s − 168.·43-s + 100.·45-s + 182.·47-s − 150.·49-s − 86.2·51-s + 37.0·53-s + 39.1·55-s + ⋯
L(s)  = 1  + 0.890·3-s − 1.59·5-s + 0.749·7-s − 0.207·9-s − 0.0599·11-s + 0.989·13-s − 1.42·15-s − 0.266·17-s − 1.47·19-s + 0.667·21-s + 1.21·23-s + 1.55·25-s − 1.07·27-s + 0.541·29-s − 0.182·31-s − 0.0533·33-s − 1.19·35-s + 0.560·37-s + 0.880·39-s + 0.801·41-s − 0.598·43-s + 0.331·45-s + 0.567·47-s − 0.438·49-s − 0.236·51-s + 0.0958·53-s + 0.0959·55-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

\(L(2)\) \(\approx\) \(2.046303324\)
\(L(\frac12)\) \(\approx\) \(2.046303324\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.62T + 27T^{2} \)
5 \( 1 + 17.8T + 125T^{2} \)
7 \( 1 - 13.8T + 343T^{2} \)
11 \( 1 + 2.18T + 1.33e3T^{2} \)
13 \( 1 - 46.3T + 2.19e3T^{2} \)
17 \( 1 + 18.6T + 4.91e3T^{2} \)
19 \( 1 + 122.T + 6.85e3T^{2} \)
23 \( 1 - 134.T + 1.21e4T^{2} \)
29 \( 1 - 84.5T + 2.43e4T^{2} \)
31 \( 1 + 31.5T + 2.97e4T^{2} \)
37 \( 1 - 126.T + 5.06e4T^{2} \)
41 \( 1 - 210.T + 6.89e4T^{2} \)
43 \( 1 + 168.T + 7.95e4T^{2} \)
47 \( 1 - 182.T + 1.03e5T^{2} \)
53 \( 1 - 37.0T + 1.48e5T^{2} \)
59 \( 1 - 624.T + 2.05e5T^{2} \)
61 \( 1 - 246.T + 2.26e5T^{2} \)
67 \( 1 + 129.T + 3.00e5T^{2} \)
71 \( 1 - 348.T + 3.57e5T^{2} \)
73 \( 1 - 299.T + 3.89e5T^{2} \)
79 \( 1 - 943.T + 4.93e5T^{2} \)
83 \( 1 - 443.T + 5.71e5T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−9.178066843917818205370677078977, −8.463710104634004924104836469355, −8.177830283690297228415335920867, −7.33284869981422888796265979773, −6.32754477513108282123565979602, −4.94030930157563842186892726486, −4.06956746066854671690371895257, −3.37349326213252338413056745894, −2.24329270540233609938183858733, −0.72215083035860880544702619973, 0.72215083035860880544702619973, 2.24329270540233609938183858733, 3.37349326213252338413056745894, 4.06956746066854671690371895257, 4.94030930157563842186892726486, 6.32754477513108282123565979602, 7.33284869981422888796265979773, 8.177830283690297228415335920867, 8.463710104634004924104836469355, 9.178066843917818205370677078977

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