Properties

Label 4-640332-1.1-c1e2-0-13
Degree $4$
Conductor $640332$
Sign $1$
Analytic cond. $40.8281$
Root an. cond. $2.52778$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 2·7-s + 9-s + 12-s + 4·13-s + 16-s + 4·19-s + 2·21-s − 10·25-s + 27-s + 2·28-s + 4·31-s + 36-s + 4·37-s + 4·39-s − 8·43-s + 48-s + 3·49-s + 4·52-s + 4·57-s + 4·61-s + 2·63-s + 64-s − 8·67-s − 8·73-s − 10·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.917·19-s + 0.436·21-s − 2·25-s + 0.192·27-s + 0.377·28-s + 0.718·31-s + 1/6·36-s + 0.657·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s + 3/7·49-s + 0.554·52-s + 0.529·57-s + 0.512·61-s + 0.251·63-s + 1/8·64-s − 0.977·67-s − 0.936·73-s − 1.15·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640332\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(40.8281\)
Root analytic conductor: \(2.52778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{640332} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640332,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.451405122\)
\(L(\frac12)\) \(\approx\) \(3.451405122\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
7$C_1$ \( ( 1 - T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.158830011188341869727925681893, −8.104024357108211588654438142125, −7.46173924472506058951788867760, −7.27269450626182225527303312385, −6.40050362102582630929450786700, −6.27935085429088375420913933534, −5.57921544774938402027851071868, −5.25821839364078131696035606890, −4.44481069307592434914952580537, −4.14440200645550446493587121840, −3.37789225057720607500704079136, −3.12443402203060754923261311981, −2.16035280583295696954900337452, −1.77631258837245971519374380176, −0.960161767014088955334438808530, 0.960161767014088955334438808530, 1.77631258837245971519374380176, 2.16035280583295696954900337452, 3.12443402203060754923261311981, 3.37789225057720607500704079136, 4.14440200645550446493587121840, 4.44481069307592434914952580537, 5.25821839364078131696035606890, 5.57921544774938402027851071868, 6.27935085429088375420913933534, 6.40050362102582630929450786700, 7.27269450626182225527303312385, 7.46173924472506058951788867760, 8.104024357108211588654438142125, 8.158830011188341869727925681893

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