Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{3} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s + 3·11-s + 2·12-s − 2·13-s − 4·16-s − 2·18-s − 6·22-s − 2·23-s + 6·25-s + 4·26-s + 27-s + 8·32-s + 3·33-s + 2·36-s − 4·37-s − 2·39-s + 6·44-s + 4·46-s − 4·47-s − 4·48-s − 10·49-s − 12·50-s − 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 0.554·13-s − 16-s − 0.471·18-s − 1.27·22-s − 0.417·23-s + 6/5·25-s + 0.784·26-s + 0.192·27-s + 1.41·32-s + 0.522·33-s + 1/3·36-s − 0.657·37-s − 0.320·39-s + 0.904·44-s + 0.589·46-s − 0.583·47-s − 0.577·48-s − 1.42·49-s − 1.69·50-s − 0.554·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4752\)    =    \(2^{4} \cdot 3^{3} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4752} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4752,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5241693756$
$L(\frac12)$  $\approx$  $0.5241693756$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( 1 - T \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.08408942282026165421113232095, −11.67246952894085626116750430085, −10.87691394832405013001401457429, −10.38936697909055843929130528429, −9.798185870861037407277263459933, −9.202343424123060901986503459506, −8.875144739305145844251978393298, −8.143283213264253094923506804290, −7.62843476028998637591464465653, −6.88163267941314471177005968060, −6.38307042519464451904685536162, −5.06221262167796761274280639428, −4.21718214290935767268143237185, −3.00611925464994524037182969368, −1.66305183534187909223413734470, 1.66305183534187909223413734470, 3.00611925464994524037182969368, 4.21718214290935767268143237185, 5.06221262167796761274280639428, 6.38307042519464451904685536162, 6.88163267941314471177005968060, 7.62843476028998637591464465653, 8.143283213264253094923506804290, 8.875144739305145844251978393298, 9.202343424123060901986503459506, 9.798185870861037407277263459933, 10.38936697909055843929130528429, 10.87691394832405013001401457429, 11.67246952894085626116750430085, 12.08408942282026165421113232095

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