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  − 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

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

\(L(2)\) \(\approx\) \(0.002746848088\)
\(L(\frac12)\) \(\approx\) \(0.002746848088\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.80T + 27T^{2} \)
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 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−9.535973382159115698684518060948, −8.957260622662167161847563934765, −7.80143013008738190081180802797, −6.83422587229596531628001942728, −6.24263744738319317874120322157, −5.33514119532206295004930501542, −4.34841080166228441311950140151, −3.10233498839197194488813361005, −2.28883260798489212004260433735, −0.02596870603313918829568212620, 0.02596870603313918829568212620, 2.28883260798489212004260433735, 3.10233498839197194488813361005, 4.34841080166228441311950140151, 5.33514119532206295004930501542, 6.24263744738319317874120322157, 6.83422587229596531628001942728, 7.80143013008738190081180802797, 8.957260622662167161847563934765, 9.535973382159115698684518060948

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