Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 19^{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 − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s − 4·11-s + 2·13-s + 14-s − 16-s + 6·17-s − 2·20-s + 4·22-s − 25-s − 2·26-s + 28-s − 2·29-s − 5·32-s − 6·34-s − 2·35-s − 6·37-s + 6·40-s + 2·41-s − 4·43-s + 4·44-s + 49-s + 50-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s − 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.883·32-s − 1.02·34-s − 0.338·35-s − 0.986·37-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.603·44-s + 1/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 22743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 22743 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22743\)    =    \(3^{2} \cdot 7 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{22743} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 22743,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.03370510653655, −15.20086476300314, −14.70250827964801, −13.88078671543458, −13.65358953082019, −13.19018578387675, −12.61006758034389, −12.05574406630107, −11.18431424540378, −10.46703071482560, −10.26725378626327, −9.663185835132306, −9.294790128672518, −8.485664246907082, −8.111657677366945, −7.479347613770856, −6.863547300305273, −5.967107539671437, −5.422330053826607, −5.113520368931750, −4.056421545440046, −3.438774026611305, −2.589864309275685, −1.764511385695925, −1.001016967903345, 0, 1.001016967903345, 1.764511385695925, 2.589864309275685, 3.438774026611305, 4.056421545440046, 5.113520368931750, 5.422330053826607, 5.967107539671437, 6.863547300305273, 7.479347613770856, 8.111657677366945, 8.485664246907082, 9.294790128672518, 9.663185835132306, 10.26725378626327, 10.46703071482560, 11.18431424540378, 12.05574406630107, 12.61006758034389, 13.19018578387675, 13.65358953082019, 13.88078671543458, 14.70250827964801, 15.20086476300314, 16.03370510653655

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