Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 23^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 5-s + 2·7-s + 2·10-s + 11-s + 4·13-s + 4·14-s − 4·16-s − 2·17-s + 2·20-s + 2·22-s − 4·25-s + 8·26-s + 4·28-s + 7·31-s − 8·32-s − 4·34-s + 2·35-s − 3·37-s + 8·41-s + 6·43-s + 2·44-s − 8·47-s − 3·49-s − 8·50-s + 8·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s + 0.755·7-s + 0.632·10-s + 0.301·11-s + 1.10·13-s + 1.06·14-s − 16-s − 0.485·17-s + 0.447·20-s + 0.426·22-s − 4/5·25-s + 1.56·26-s + 0.755·28-s + 1.25·31-s − 1.41·32-s − 0.685·34-s + 0.338·35-s − 0.493·37-s + 1.24·41-s + 0.914·43-s + 0.301·44-s − 1.16·47-s − 3/7·49-s − 1.13·50-s + 1.10·52-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  =  \(0\)
Selberg data  =  \((2,\ 52371,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(6.875768017\)
\(L(\frac12)\)  \(\approx\)  \(6.875768017\)
\(L(\frac{3}{2})\)   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 \( 1 - p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.31465818503998, −13.88176445543968, −13.61651365617033, −13.04590340420160, −12.53577105115097, −12.04845951285428, −11.36243189859812, −11.19466583085665, −10.59382494906951, −9.829088148410501, −9.221142799404981, −8.789011567064715, −8.066676450133041, −7.630078979408841, −6.710637091222503, −6.215962534163355, −5.985643413555836, −5.228176311617369, −4.631219267187221, −4.306078861879373, −3.537622328306707, −3.061814718862568, −2.159540069378382, −1.703866975198364, −0.7134147154498627, 0.7134147154498627, 1.703866975198364, 2.159540069378382, 3.061814718862568, 3.537622328306707, 4.306078861879373, 4.631219267187221, 5.228176311617369, 5.985643413555836, 6.215962534163355, 6.710637091222503, 7.630078979408841, 8.066676450133041, 8.789011567064715, 9.221142799404981, 9.829088148410501, 10.59382494906951, 11.19466583085665, 11.36243189859812, 12.04845951285428, 12.53577105115097, 13.04590340420160, 13.61651365617033, 13.88176445543968, 14.31465818503998

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