Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{2} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 6·11-s − 4·13-s + 4·19-s − 6·23-s + 5·25-s + 27-s + 12·29-s − 8·31-s + 6·33-s − 2·37-s + 4·39-s − 24·41-s + 8·43-s − 12·47-s + 6·53-s − 4·57-s − 10·61-s + 8·67-s + 6·69-s − 12·71-s − 10·73-s − 5·75-s − 4·79-s − 81-s − 24·83-s − 12·87-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.80·11-s − 1.10·13-s + 0.917·19-s − 1.25·23-s + 25-s + 0.192·27-s + 2.22·29-s − 1.43·31-s + 1.04·33-s − 0.328·37-s + 0.640·39-s − 3.74·41-s + 1.21·43-s − 1.75·47-s + 0.824·53-s − 0.529·57-s − 1.28·61-s + 0.977·67-s + 0.722·69-s − 1.42·71-s − 1.17·73-s − 0.577·75-s − 0.450·79-s − 1/9·81-s − 2.63·83-s − 1.28·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2352} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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

−8.646831787816903824132726586090, −8.412982407572827495968691533416, −8.001374879259783565186792039005, −7.68051665998343196354861182960, −7.07481749841635427497603873488, −7.05600172023507380875831306940, −6.48153846101342545074534177586, −6.01013707793015992559131022600, −5.51503449841269118031303061949, −5.19296748933218413911097647765, −4.90656848857707710336417235796, −4.65757354639730831547858843281, −3.93870317497580443598766352268, −3.33063664836859835367763984276, −2.80677800842291093127547084844, −2.66195659905419859346654672789, −1.83020412214016604043834328167, −1.27726399541645043448534805157, 0, 0, 1.27726399541645043448534805157, 1.83020412214016604043834328167, 2.66195659905419859346654672789, 2.80677800842291093127547084844, 3.33063664836859835367763984276, 3.93870317497580443598766352268, 4.65757354639730831547858843281, 4.90656848857707710336417235796, 5.19296748933218413911097647765, 5.51503449841269118031303061949, 6.01013707793015992559131022600, 6.48153846101342545074534177586, 7.05600172023507380875831306940, 7.07481749841635427497603873488, 7.68051665998343196354861182960, 8.001374879259783565186792039005, 8.412982407572827495968691533416, 8.646831787816903824132726586090

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