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-s − 4-s + 2·7-s − 3·8-s + 11-s + 2·13-s + 2·14-s − 16-s − 2·19-s + 22-s − 5·25-s + 2·26-s − 2·28-s + 10·29-s + 4·31-s + 5·32-s − 2·37-s − 2·38-s + 2·41-s − 2·43-s − 44-s + 8·47-s − 3·49-s − 5·50-s − 2·52-s − 4·53-s − 6·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s + 0.301·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.458·19-s + 0.213·22-s − 25-s + 0.392·26-s − 0.377·28-s + 1.85·29-s + 0.718·31-s + 0.883·32-s − 0.328·37-s − 0.324·38-s + 0.312·41-s − 0.304·43-s − 0.150·44-s + 1.16·47-s − 3/7·49-s − 0.707·50-s − 0.277·52-s − 0.549·53-s − 0.801·56-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  =  \((2,\ 52371,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.102258896\)
\(L(\frac12)\)  \(\approx\)  \(3.102258896\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.51734749173911, −13.89053611255140, −13.56954938786448, −13.05471765109687, −12.44430993090102, −11.91071976104484, −11.64102641962457, −10.96440247044779, −10.32469183414122, −9.872946501184478, −9.219898913644485, −8.589604390638961, −8.310875102957904, −7.779339333779191, −6.838602113906760, −6.438157936661531, −5.753060143686566, −5.326134703752708, −4.541181074382932, −4.298284439965390, −3.642034132395196, −2.934771135440209, −2.240610524163384, −1.341045648258519, −0.5860099098897066, 0.5860099098897066, 1.341045648258519, 2.240610524163384, 2.934771135440209, 3.642034132395196, 4.298284439965390, 4.541181074382932, 5.326134703752708, 5.753060143686566, 6.438157936661531, 6.838602113906760, 7.779339333779191, 8.310875102957904, 8.589604390638961, 9.219898913644485, 9.872946501184478, 10.32469183414122, 10.96440247044779, 11.64102641962457, 11.91071976104484, 12.44430993090102, 13.05471765109687, 13.56954938786448, 13.89053611255140, 14.51734749173911

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