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  − 7.77·3-s − 6.59·5-s + 24.8·7-s + 33.4·9-s − 31.5·11-s − 15.9·13-s + 51.2·15-s + 88.4·17-s + 53.4·19-s − 193.·21-s + 48.1·23-s − 81.4·25-s − 49.8·27-s − 14.7·29-s − 96.9·31-s + 245.·33-s − 164.·35-s − 230.·37-s + 123.·39-s − 360.·41-s + 141.·43-s − 220.·45-s − 220.·47-s + 276.·49-s − 687.·51-s + 248.·53-s + 208.·55-s + ⋯
L(s)  = 1  − 1.49·3-s − 0.589·5-s + 1.34·7-s + 1.23·9-s − 0.866·11-s − 0.340·13-s + 0.882·15-s + 1.26·17-s + 0.645·19-s − 2.01·21-s + 0.436·23-s − 0.651·25-s − 0.355·27-s − 0.0945·29-s − 0.561·31-s + 1.29·33-s − 0.793·35-s − 1.02·37-s + 0.508·39-s − 1.37·41-s + 0.502·43-s − 0.730·45-s − 0.684·47-s + 0.807·49-s − 1.88·51-s + 0.644·53-s + 0.510·55-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

\(L(2)\) \(\approx\) \(0.9504992470\)
\(L(\frac12)\) \(\approx\) \(0.9504992470\)
\(L(\frac{5}{2})\) 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 \( 1 + 7.77T + 27T^{2} \)
5 \( 1 + 6.59T + 125T^{2} \)
7 \( 1 - 24.8T + 343T^{2} \)
11 \( 1 + 31.5T + 1.33e3T^{2} \)
13 \( 1 + 15.9T + 2.19e3T^{2} \)
17 \( 1 - 88.4T + 4.91e3T^{2} \)
19 \( 1 - 53.4T + 6.85e3T^{2} \)
23 \( 1 - 48.1T + 1.21e4T^{2} \)
29 \( 1 + 14.7T + 2.43e4T^{2} \)
31 \( 1 + 96.9T + 2.97e4T^{2} \)
37 \( 1 + 230.T + 5.06e4T^{2} \)
41 \( 1 + 360.T + 6.89e4T^{2} \)
43 \( 1 - 141.T + 7.95e4T^{2} \)
47 \( 1 + 220.T + 1.03e5T^{2} \)
53 \( 1 - 248.T + 1.48e5T^{2} \)
59 \( 1 - 572.T + 2.05e5T^{2} \)
61 \( 1 + 939.T + 2.26e5T^{2} \)
67 \( 1 - 151.T + 3.00e5T^{2} \)
71 \( 1 + 215.T + 3.57e5T^{2} \)
73 \( 1 + 668.T + 3.89e5T^{2} \)
79 \( 1 - 822.T + 4.93e5T^{2} \)
83 \( 1 - 462.T + 5.71e5T^{2} \)
89 \( 1 - 262.T + 7.04e5T^{2} \)
97 \( 1 + 150.T + 9.12e5T^{2} \)
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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

−9.899916839396347617259534433390, −8.547492282684873001145166849485, −7.68395373996764815859446694777, −7.20010396315149992328578725550, −5.86647251173344447252028226328, −5.19720241949501656083139757098, −4.71109155714413677071267459567, −3.39408373228466525539790772773, −1.72875358780219454224521890928, −0.57239065623834175270607073818, 0.57239065623834175270607073818, 1.72875358780219454224521890928, 3.39408373228466525539790772773, 4.71109155714413677071267459567, 5.19720241949501656083139757098, 5.86647251173344447252028226328, 7.20010396315149992328578725550, 7.68395373996764815859446694777, 8.547492282684873001145166849485, 9.899916839396347617259534433390

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