Properties

Degree 2
Conductor $ 2^{10} $
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

\( d \)  =  \(2\)
\( N \)  =  \(1024\)    =    \(2^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{1024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1024,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(5.362204327\)
\(L(\frac12)\)  \(\approx\)  \(5.362204327\)
\(L(\frac{5}{2})\)   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 \( 1 - 7.77T + 27T^{2} \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.447676313893274432504383523748, −8.619683753003641797333693601405, −8.112563035487685886760342712753, −7.34955602402903928012080816777, −6.19892573465474187681605278880, −5.11638580626727951862460017680, −4.07413289258415650053973837289, −3.18853266757187638752687406080, −1.97656705322758935871948669161, −1.37068447099714985526463778394, 1.37068447099714985526463778394, 1.97656705322758935871948669161, 3.18853266757187638752687406080, 4.07413289258415650053973837289, 5.11638580626727951862460017680, 6.19892573465474187681605278880, 7.34955602402903928012080816777, 8.112563035487685886760342712753, 8.619683753003641797333693601405, 9.447676313893274432504383523748

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