Properties

Label 4-672e2-1.1-c1e2-0-81
Degree $4$
Conductor $451584$
Sign $1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 9-s − 8·13-s − 12·19-s + 4·21-s − 10·25-s + 4·27-s − 8·31-s + 20·37-s + 16·39-s + 8·43-s + 3·49-s + 24·57-s − 16·61-s − 2·63-s − 16·67-s − 12·73-s + 20·75-s − 32·79-s − 11·81-s + 16·91-s + 16·93-s − 4·97-s − 8·103-s + 20·109-s − 40·111-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s − 2.21·13-s − 2.75·19-s + 0.872·21-s − 2·25-s + 0.769·27-s − 1.43·31-s + 3.28·37-s + 2.56·39-s + 1.21·43-s + 3/7·49-s + 3.17·57-s − 2.04·61-s − 0.251·63-s − 1.95·67-s − 1.40·73-s + 2.30·75-s − 3.60·79-s − 1.22·81-s + 1.67·91-s + 1.65·93-s − 0.406·97-s − 0.788·103-s + 1.91·109-s − 3.79·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(28.7933\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−7.85643204119898614530616966999, −7.56688031432348988896561867945, −7.31000008746359385531868028906, −6.52662751327895896284855360684, −6.17236708003451018025878030760, −5.88296122377187582734081048905, −5.48532390305501388420276275059, −4.57048039337710861640747663017, −4.39654298826148920136877245563, −3.99709320456544753258230397198, −2.70117202411251547524653201529, −2.62612435759905589929421296030, −1.69348648530400294148534200114, 0, 0, 1.69348648530400294148534200114, 2.62612435759905589929421296030, 2.70117202411251547524653201529, 3.99709320456544753258230397198, 4.39654298826148920136877245563, 4.57048039337710861640747663017, 5.48532390305501388420276275059, 5.88296122377187582734081048905, 6.17236708003451018025878030760, 6.52662751327895896284855360684, 7.31000008746359385531868028906, 7.56688031432348988896561867945, 7.85643204119898614530616966999

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