Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \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 − 4-s + 4·5-s + 2·7-s + 3·8-s − 4·10-s + 11-s − 2·13-s − 2·14-s − 16-s − 2·17-s + 6·19-s − 4·20-s − 22-s + 11·25-s + 2·26-s − 2·28-s − 6·29-s + 4·31-s − 5·32-s + 2·34-s + 8·35-s + 6·37-s − 6·38-s + 12·40-s − 10·41-s − 6·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.78·5-s + 0.755·7-s + 1.06·8-s − 1.26·10-s + 0.301·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.894·20-s − 0.213·22-s + 11/5·25-s + 0.392·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 1.35·35-s + 0.986·37-s − 0.973·38-s + 1.89·40-s − 1.56·41-s − 0.914·43-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  =  \(1\)
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 \( 1 - 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.58896177992500, −14.24249620300196, −13.54238052080117, −13.38129582194344, −12.97765473326721, −12.07711688289325, −11.57924368052024, −10.97340958331146, −10.42010217012633, −9.778569599722382, −9.639150460791499, −9.234951916144144, −8.439192003706032, −8.182280024875207, −7.366812183911304, −6.861886855934621, −6.240673985105784, −5.460738621083916, −5.113087515573725, −4.701851235233339, −3.829620838541927, −2.948522116723307, −2.217465608749423, −1.500214486549941, −1.233301204848757, 0, 1.233301204848757, 1.500214486549941, 2.217465608749423, 2.948522116723307, 3.829620838541927, 4.701851235233339, 5.113087515573725, 5.460738621083916, 6.240673985105784, 6.861886855934621, 7.366812183911304, 8.182280024875207, 8.439192003706032, 9.234951916144144, 9.639150460791499, 9.778569599722382, 10.42010217012633, 10.97340958331146, 11.57924368052024, 12.07711688289325, 12.97765473326721, 13.38129582194344, 13.54238052080117, 14.24249620300196, 14.58896177992500

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