Properties

Degree 4
Conductor $ 3^{3} \cdot 7^{2} \cdot 19^{2} $
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  + 2·2-s − 3-s − 4-s − 2·6-s − 2·7-s − 8·8-s + 9-s + 12-s − 4·14-s − 7·16-s + 2·18-s + 4·19-s + 2·21-s + 8·24-s − 6·25-s − 27-s + 2·28-s + 4·29-s + 14·32-s − 36-s + 8·38-s − 4·41-s + 4·42-s − 8·43-s + 7·48-s + 3·49-s − 12·50-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s − 1/2·4-s − 0.816·6-s − 0.755·7-s − 2.82·8-s + 1/3·9-s + 0.288·12-s − 1.06·14-s − 7/4·16-s + 0.471·18-s + 0.917·19-s + 0.436·21-s + 1.63·24-s − 6/5·25-s − 0.192·27-s + 0.377·28-s + 0.742·29-s + 2.47·32-s − 1/6·36-s + 1.29·38-s − 0.624·41-s + 0.617·42-s − 1.21·43-s + 1.01·48-s + 3/7·49-s − 1.69·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(477603\)    =    \(3^{3} \cdot 7^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{477603} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 477603,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.9674071593\)
\(L(\frac12)\)  \(\approx\)  \(0.9674071593\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( 1 + T \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−8.694062888305852644281305198602, −7.85302053965317674139971119169, −7.81295430367747747098376616892, −6.87477517530779045173647189781, −6.42842960047713503559988653252, −5.94607665445287604157300008429, −5.82904766004805566557712895862, −5.02709416296405066539370067618, −4.81936947701198622186082780840, −4.36029050874640221146718646433, −3.73298077731310404370734463266, −3.13006637509653258681236844743, −3.05422074105458389777226041971, −1.71320545529150591858508044629, −0.43525041789359895883868375518, 0.43525041789359895883868375518, 1.71320545529150591858508044629, 3.05422074105458389777226041971, 3.13006637509653258681236844743, 3.73298077731310404370734463266, 4.36029050874640221146718646433, 4.81936947701198622186082780840, 5.02709416296405066539370067618, 5.82904766004805566557712895862, 5.94607665445287604157300008429, 6.42842960047713503559988653252, 6.87477517530779045173647189781, 7.81295430367747747098376616892, 7.85302053965317674139971119169, 8.694062888305852644281305198602

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