Properties

Degree 4
Conductor $ 7^{2} \cdot 11^{3} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 3·4-s − 4·5-s + 6·9-s + 11-s − 12·12-s − 16·15-s + 5·16-s + 12·20-s − 8·23-s + 2·25-s − 4·27-s + 20·31-s + 4·33-s − 18·36-s − 12·37-s − 3·44-s − 24·45-s − 20·47-s + 20·48-s + 49-s − 12·53-s − 4·55-s + 4·59-s + 48·60-s − 3·64-s + 16·67-s + ⋯
L(s)  = 1  + 2.30·3-s − 3/2·4-s − 1.78·5-s + 2·9-s + 0.301·11-s − 3.46·12-s − 4.13·15-s + 5/4·16-s + 2.68·20-s − 1.66·23-s + 2/5·25-s − 0.769·27-s + 3.59·31-s + 0.696·33-s − 3·36-s − 1.97·37-s − 0.452·44-s − 3.57·45-s − 2.91·47-s + 2.88·48-s + 1/7·49-s − 1.64·53-s − 0.539·55-s + 0.520·59-s + 6.19·60-s − 3/8·64-s + 1.95·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(65219\)    =    \(7^{2} \cdot 11^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{65219} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 65219,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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

−9.742618950410792459548371471551, −8.678915900547504955372461266927, −8.572504027942197515704035369998, −8.339912824949481407935374648788, −7.82867506007281568614416512666, −7.67378178112872223010860918103, −6.70583207072298392541889211635, −5.96443978110022033983848919969, −4.94559945005359926667122742610, −4.28286203006294711588571475853, −4.07930883034575373640745113861, −3.26888344602981049454952325554, −3.17103530919462504679231815837, −1.91939450772296709675161614006, 0, 1.91939450772296709675161614006, 3.17103530919462504679231815837, 3.26888344602981049454952325554, 4.07930883034575373640745113861, 4.28286203006294711588571475853, 4.94559945005359926667122742610, 5.96443978110022033983848919969, 6.70583207072298392541889211635, 7.67378178112872223010860918103, 7.82867506007281568614416512666, 8.339912824949481407935374648788, 8.572504027942197515704035369998, 8.678915900547504955372461266927, 9.742618950410792459548371471551

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