Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 3·9-s − 16·19-s + 2·25-s + 12·29-s + 8·43-s − 12·45-s + 16·47-s + 49-s + 12·53-s + 8·67-s + 16·71-s + 20·73-s + 9·81-s − 64·95-s − 12·97-s + 4·101-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 1.78·5-s − 9-s − 3.67·19-s + 2/5·25-s + 2.22·29-s + 1.21·43-s − 1.78·45-s + 2.33·47-s + 1/7·49-s + 1.64·53-s + 0.977·67-s + 1.89·71-s + 2.34·73-s + 81-s − 6.56·95-s − 1.21·97-s + 0.398·101-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-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

\( d \)  =  \(4\)
\( N \)  =  \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{451584} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 451584,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.152524652$
$L(\frac12)$  $\approx$  $2.152524652$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.560607806232820418165457845354, −8.272203299109375537884659130698, −7.902596514850714806179317231692, −6.84383926850302326577754552348, −6.67949194679031572918581545842, −6.20669276463558444353009673367, −5.82613210142107824134972732669, −5.50140758689236696304146355325, −4.83297349612888492276603651901, −4.18465932715971087267449515138, −3.83348667018088193321396729321, −2.66451378084199854982327930143, −2.24055198785635289229441777506, −2.14556791802447766709825303506, −0.76540944404158607646425160976, 0.76540944404158607646425160976, 2.14556791802447766709825303506, 2.24055198785635289229441777506, 2.66451378084199854982327930143, 3.83348667018088193321396729321, 4.18465932715971087267449515138, 4.83297349612888492276603651901, 5.50140758689236696304146355325, 5.82613210142107824134972732669, 6.20669276463558444353009673367, 6.67949194679031572918581545842, 6.84383926850302326577754552348, 7.902596514850714806179317231692, 8.272203299109375537884659130698, 8.560607806232820418165457845354

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