Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 23^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·5-s − 4·7-s + 3·8-s + 2·10-s + 11-s − 2·13-s + 4·14-s − 16-s − 2·17-s + 2·20-s − 22-s − 25-s + 2·26-s + 4·28-s + 6·29-s − 8·31-s − 5·32-s + 2·34-s + 8·35-s − 6·37-s − 6·40-s + 2·41-s − 44-s − 8·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 1.06·8-s + 0.632·10-s + 0.301·11-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.447·20-s − 0.213·22-s − 1/5·25-s + 0.392·26-s + 0.755·28-s + 1.11·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s + 1.35·35-s − 0.986·37-s − 0.948·40-s + 0.312·41-s − 0.150·44-s − 1.16·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(52371\)    =    \(3^{2} \cdot 11 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{52371} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 52371,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;23\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 - T \)
23 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.94378469122818, −14.60426983405236, −13.81148843868324, −13.44152298757676, −12.92174818263866, −12.37119828575122, −12.04285207235631, −11.31644794199367, −10.72782710029507, −10.21147568430030, −9.738981670882215, −9.272745044527853, −8.818248143801533, −8.277954733007426, −7.685750970948436, −7.085213148766898, −6.773968303503365, −6.021046403613920, −5.286516650045761, −4.641011094063604, −3.973762985853764, −3.590595170353258, −2.902907774293462, −2.030192426721872, −1.069744467268671, 0, 0, 1.069744467268671, 2.030192426721872, 2.902907774293462, 3.590595170353258, 3.973762985853764, 4.641011094063604, 5.286516650045761, 6.021046403613920, 6.773968303503365, 7.085213148766898, 7.685750970948436, 8.277954733007426, 8.818248143801533, 9.272745044527853, 9.738981670882215, 10.21147568430030, 10.72782710029507, 11.31644794199367, 12.04285207235631, 12.37119828575122, 12.92174818263866, 13.44152298757676, 13.81148843868324, 14.60426983405236, 14.94378469122818

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