Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{3} \cdot 11^{3} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 11-s − 8·17-s − 6·25-s − 27-s + 33-s − 12·37-s − 10·49-s + 8·51-s + 8·67-s + 6·75-s + 81-s + 32·83-s − 4·97-s − 99-s + 16·101-s − 24·103-s + 32·107-s + 12·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 10·147-s + 149-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.94·17-s − 6/5·25-s − 0.192·27-s + 0.174·33-s − 1.97·37-s − 1.42·49-s + 1.12·51-s + 0.977·67-s + 0.692·75-s + 1/9·81-s + 3.51·83-s − 0.406·97-s − 0.100·99-s + 1.59·101-s − 2.36·103-s + 3.09·107-s + 1.13·111-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.824·147-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 574992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(574992\)    =    \(2^{4} \cdot 3^{3} \cdot 11^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{574992} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 574992,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.8340465365$
$L(\frac12)$  $\approx$  $0.8340465365$
$L(\frac{3}{2})$   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 \( 1 \)
3$C_1$ \( 1 + T \)
11$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.363783361006180122238656454062, −8.017674089261979349140094913788, −7.49718153606969186728697393667, −7.00719281751835795853753965763, −6.41893627610765861044252902088, −6.41512539021783853545914554633, −5.62860857478680063008223727075, −5.12414628938185870335019819699, −4.78818744575862870088978876704, −4.18020563951979381603420573914, −3.67380578777293619845336752383, −3.05194549653743224577997918283, −2.07022010244043316224023560102, −1.86785039249087723872477729694, −0.46511289287587154532633948638, 0.46511289287587154532633948638, 1.86785039249087723872477729694, 2.07022010244043316224023560102, 3.05194549653743224577997918283, 3.67380578777293619845336752383, 4.18020563951979381603420573914, 4.78818744575862870088978876704, 5.12414628938185870335019819699, 5.62860857478680063008223727075, 6.41512539021783853545914554633, 6.41893627610765861044252902088, 7.00719281751835795853753965763, 7.49718153606969186728697393667, 8.017674089261979349140094913788, 8.363783361006180122238656454062

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