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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s + 11-s − 2·13-s − 16-s + 2·17-s + 4·19-s + 2·20-s + 22-s − 25-s − 2·26-s + 2·29-s + 5·32-s + 2·34-s + 10·37-s + 4·38-s + 6·40-s + 6·41-s + 12·43-s − 44-s − 7·49-s − 50-s + 2·52-s + 14·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 0.554·13-s − 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s − 0.392·26-s + 0.371·29-s + 0.883·32-s + 0.342·34-s + 1.64·37-s + 0.648·38-s + 0.948·40-s + 0.937·41-s + 1.82·43-s − 0.150·44-s − 49-s − 0.141·50-s + 0.277·52-s + 1.92·53-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  =  \((2,\ 52371,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.105779405\)
\(L(\frac12)\)  \(\approx\)  \(2.105779405\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.38628846052709, −14.07575280622064, −13.49538354593320, −12.79110380461936, −12.57641391648440, −11.92412650562716, −11.58748768516109, −11.13703629531450, −10.28257826858492, −9.725807145027415, −9.335696125734861, −8.742798499720485, −8.089549445871269, −7.606705885196776, −7.197771119856228, −6.314386076665847, −5.806173250791369, −5.268744028907587, −4.609947587857553, −4.081072158651704, −3.733705681627742, −2.908340234032158, −2.468736990381569, −1.146827184503603, −0.5229780054106657, 0.5229780054106657, 1.146827184503603, 2.468736990381569, 2.908340234032158, 3.733705681627742, 4.081072158651704, 4.609947587857553, 5.268744028907587, 5.806173250791369, 6.314386076665847, 7.197771119856228, 7.606705885196776, 8.089549445871269, 8.742798499720485, 9.335696125734861, 9.725807145027415, 10.28257826858492, 11.13703629531450, 11.58748768516109, 11.92412650562716, 12.57641391648440, 12.79110380461936, 13.49538354593320, 14.07575280622064, 14.38628846052709

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