Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 23^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 2·12-s + 4·13-s + 16-s + 6·17-s − 18-s + 2·19-s − 2·24-s − 5·25-s − 4·26-s − 4·27-s − 6·29-s + 4·31-s − 32-s − 6·34-s + 36-s − 2·37-s − 2·38-s + 8·39-s − 6·41-s − 8·43-s + 12·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 51842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(51842\)    =    \(2 \cdot 7^{2} \cdot 23^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{51842} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 51842,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.71688946650247, −14.20320025609894, −13.83501630963758, −13.31033589441407, −12.81926668702262, −12.01878168502246, −11.62333605457552, −11.17705205494114, −10.32275486653526, −10.03890120975865, −9.436254157215824, −9.022458906380164, −8.354753202648190, −8.110149332059267, −7.597697606542388, −7.010624023945231, −6.333816121314228, −5.571323083846240, −5.326160933068359, −3.974482093861297, −3.739191873824977, −3.068494989002011, −2.495778332910390, −1.637430993271355, −1.181738165349491, 0, 1.181738165349491, 1.637430993271355, 2.495778332910390, 3.068494989002011, 3.739191873824977, 3.974482093861297, 5.326160933068359, 5.571323083846240, 6.333816121314228, 7.010624023945231, 7.597697606542388, 8.110149332059267, 8.354753202648190, 9.022458906380164, 9.436254157215824, 10.03890120975865, 10.32275486653526, 11.17705205494114, 11.62333605457552, 12.01878168502246, 12.81926668702262, 13.31033589441407, 13.83501630963758, 14.20320025609894, 14.71688946650247

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