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-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 3·11-s − 12-s + 4·13-s − 16-s + 18-s − 3·22-s − 8·23-s − 3·24-s − 6·25-s + 4·26-s + 27-s + 5·32-s − 3·33-s − 36-s − 4·37-s + 4·39-s + 3·44-s − 8·46-s + 8·47-s − 48-s + 2·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s − 1/4·16-s + 0.235·18-s − 0.639·22-s − 1.66·23-s − 0.612·24-s − 6/5·25-s + 0.784·26-s + 0.192·27-s + 0.883·32-s − 0.522·33-s − 1/6·36-s − 0.657·37-s + 0.640·39-s + 0.452·44-s − 1.17·46-s + 1.16·47-s − 0.144·48-s + 2/7·49-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  =  $(4,\ 4752,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.115459211$
$L(\frac12)$  $\approx$  $1.115459211$
$L(\frac{3}{2})$   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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 - T \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−12.40034919641662132775971763573, −11.94079317736327500240864866855, −11.24926867346879304788630911023, −10.43264856980990269535257796694, −9.975026179182169205209018836422, −9.304691211808039522291402854248, −8.601379066356035322593763260494, −8.146840241884356568250815540150, −7.53321652078339631746735295713, −6.43759152388885102986219540336, −5.80118775057511944644355096785, −5.13576111489458385632317052281, −4.03826779880850438115451415591, −3.63561346253081326411157377370, −2.35238971472037862013416227689, 2.35238971472037862013416227689, 3.63561346253081326411157377370, 4.03826779880850438115451415591, 5.13576111489458385632317052281, 5.80118775057511944644355096785, 6.43759152388885102986219540336, 7.53321652078339631746735295713, 8.146840241884356568250815540150, 8.601379066356035322593763260494, 9.304691211808039522291402854248, 9.975026179182169205209018836422, 10.43264856980990269535257796694, 11.24926867346879304788630911023, 11.94079317736327500240864866855, 12.40034919641662132775971763573

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