Properties

Degree $4$
Conductor $451584$
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  + 2·3-s + 3·9-s + 8·11-s + 4·17-s + 8·19-s − 6·25-s + 4·27-s + 16·33-s − 12·41-s + 8·43-s + 49-s + 8·51-s + 16·57-s − 8·59-s − 8·67-s + 20·73-s − 12·75-s + 5·81-s + 8·83-s − 12·89-s − 28·97-s + 24·99-s − 24·107-s − 28·113-s + 26·121-s − 24·123-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 2.41·11-s + 0.970·17-s + 1.83·19-s − 6/5·25-s + 0.769·27-s + 2.78·33-s − 1.87·41-s + 1.21·43-s + 1/7·49-s + 1.12·51-s + 2.11·57-s − 1.04·59-s − 0.977·67-s + 2.34·73-s − 1.38·75-s + 5/9·81-s + 0.878·83-s − 1.27·89-s − 2.84·97-s + 2.41·99-s − 2.32·107-s − 2.63·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-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: $1$
Motivic weight: \(1\)
Character: $\chi_{451584} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.875464647\)
\(L(\frac12)\) \(\approx\) \(3.875464647\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.579197535720292899748335278003, −7.963562405403294064353399542187, −7.941030481552254497163330431349, −7.14585602865319249961807679325, −6.89499765960031010222506198709, −6.43557747993001049134741297977, −5.69761771888975023011122238972, −5.40100953109829334895217907278, −4.54031360247962128301261767712, −4.04752481141371955267217260485, −3.54052974946578322805924483948, −3.29683272363669349411451437037, −2.47888758091438588042078740356, −1.50800619952171259085588630937, −1.23111713389498587746226154587, 1.23111713389498587746226154587, 1.50800619952171259085588630937, 2.47888758091438588042078740356, 3.29683272363669349411451437037, 3.54052974946578322805924483948, 4.04752481141371955267217260485, 4.54031360247962128301261767712, 5.40100953109829334895217907278, 5.69761771888975023011122238972, 6.43557747993001049134741297977, 6.89499765960031010222506198709, 7.14585602865319249961807679325, 7.941030481552254497163330431349, 7.963562405403294064353399542187, 8.579197535720292899748335278003

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