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  − 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

\( d \)  =  \(2\)
\( N \)  =  \(1024\)    =    \(2^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{1024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1024,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(2.207526728\)
\(L(\frac12)\)  \(\approx\)  \(2.207526728\)
\(L(\frac{5}{2})\)   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 \( 1 + 4.62T + 27T^{2} \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.481495775563623732639962675612, −9.100174846290486424359952704520, −7.76105912531889237885744736693, −6.87620716681115594132372500429, −5.95473055073803168537629884883, −5.27133530803626868856683901837, −4.81487967790560207153667663445, −2.99304256882878182906510455231, −1.95025969989089373048976509320, −0.848821938234964779563975377470, 0.848821938234964779563975377470, 1.95025969989089373048976509320, 2.99304256882878182906510455231, 4.81487967790560207153667663445, 5.27133530803626868856683901837, 5.95473055073803168537629884883, 6.87620716681115594132372500429, 7.76105912531889237885744736693, 9.100174846290486424359952704520, 9.481495775563623732639962675612

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