Properties

Degree 2
Conductor $ 2^{2} \cdot 6571 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s − 5·7-s + 9-s − 6·11-s − 6·13-s + 4·15-s − 3·17-s − 8·19-s + 10·21-s − 2·23-s − 25-s + 4·27-s + 2·29-s − 9·31-s + 12·33-s + 10·35-s − 4·37-s + 12·39-s − 6·41-s − 12·43-s − 2·45-s + 3·47-s + 18·49-s + 6·51-s + 3·53-s + 12·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 1.88·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s + 1.03·15-s − 0.727·17-s − 1.83·19-s + 2.18·21-s − 0.417·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s − 1.61·31-s + 2.08·33-s + 1.69·35-s − 0.657·37-s + 1.92·39-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 0.437·47-s + 18/7·49-s + 0.840·51-s + 0.412·53-s + 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(26284\)    =    \(2^{2} \cdot 6571\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{26284} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(3\)
Selberg data  =  \((2,\ 26284,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;6571\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;6571\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
6571 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−16.28783902585403, −15.63085519753034, −15.17974471448100, −14.85108039787864, −13.76617892649391, −13.14633390513072, −12.69938646170797, −12.51391685969301, −11.81863384711516, −11.36680393471989, −10.48808977766369, −10.28181409061573, −9.950524865024282, −8.898408694194140, −8.504891867067960, −7.620618398444810, −7.061695267555376, −6.730112756797060, −5.942049684660046, −5.482543017149340, −4.783586214425482, −4.223301619979378, −3.366145361083828, −2.712816972926759, −2.083287004050114, 0, 0, 0, 2.083287004050114, 2.712816972926759, 3.366145361083828, 4.223301619979378, 4.783586214425482, 5.482543017149340, 5.942049684660046, 6.730112756797060, 7.061695267555376, 7.620618398444810, 8.504891867067960, 8.898408694194140, 9.950524865024282, 10.28181409061573, 10.48808977766369, 11.36680393471989, 11.81863384711516, 12.51391685969301, 12.69938646170797, 13.14633390513072, 13.76617892649391, 14.85108039787864, 15.17974471448100, 15.63085519753034, 16.28783902585403

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