Properties

Degree 4
Conductor $ 2^{6} \cdot 7^{2} \cdot 11^{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-s + 4·3-s − 4-s + 4·6-s − 3·8-s + 6·9-s + 2·11-s − 4·12-s − 16-s + 8·17-s + 6·18-s + 2·22-s − 12·24-s − 6·25-s − 4·27-s + 5·32-s + 8·33-s + 8·34-s − 6·36-s + 8·41-s + 24·43-s − 2·44-s − 4·48-s + 49-s − 6·50-s + 32·51-s − 4·54-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s − 1/2·4-s + 1.63·6-s − 1.06·8-s + 2·9-s + 0.603·11-s − 1.15·12-s − 1/4·16-s + 1.94·17-s + 1.41·18-s + 0.426·22-s − 2.44·24-s − 6/5·25-s − 0.769·27-s + 0.883·32-s + 1.39·33-s + 1.37·34-s − 36-s + 1.24·41-s + 3.65·43-s − 0.301·44-s − 0.577·48-s + 1/7·49-s − 0.848·50-s + 4.48·51-s − 0.544·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(379456\)    =    \(2^{6} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{379456} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 379456,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(5.039087435\)
\(L(\frac12)\)  \(\approx\)  \(5.039087435\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.678915900547504955372461266927, −8.318055770167249209020787556529, −7.82867506007281568614416512666, −7.50242612035465341765985002833, −7.15095220920519743848333951211, −5.96443978110022033983848919969, −5.89346819607895550605187044438, −5.47251856550031920805970764164, −4.46493777161093979419971313423, −4.07930883034575373640745113861, −3.61703962519106612479648235899, −3.26888344602981049454952325554, −2.61963574407090955942422458621, −2.22212830718058091155401288728, −1.05751961479654851181720694519, 1.05751961479654851181720694519, 2.22212830718058091155401288728, 2.61963574407090955942422458621, 3.26888344602981049454952325554, 3.61703962519106612479648235899, 4.07930883034575373640745113861, 4.46493777161093979419971313423, 5.47251856550031920805970764164, 5.89346819607895550605187044438, 5.96443978110022033983848919969, 7.15095220920519743848333951211, 7.50242612035465341765985002833, 7.82867506007281568614416512666, 8.318055770167249209020787556529, 8.678915900547504955372461266927

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