Properties

Label 4-1632e2-1.1-c1e2-0-48
Degree $4$
Conductor $2663424$
Sign $-1$
Analytic cond. $169.822$
Root an. cond. $3.60992$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s − 2·17-s − 8·19-s + 6·25-s + 4·27-s − 20·41-s + 8·43-s − 10·49-s − 4·51-s − 16·57-s − 24·59-s + 24·67-s + 4·73-s + 12·75-s + 5·81-s + 24·83-s − 4·89-s + 12·97-s + 4·113-s − 22·121-s − 40·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 0.485·17-s − 1.83·19-s + 6/5·25-s + 0.769·27-s − 3.12·41-s + 1.21·43-s − 1.42·49-s − 0.560·51-s − 2.11·57-s − 3.12·59-s + 2.93·67-s + 0.468·73-s + 1.38·75-s + 5/9·81-s + 2.63·83-s − 0.423·89-s + 1.21·97-s + 0.376·113-s − 2·121-s − 3.60·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2663424\)    =    \(2^{10} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(169.822\)
Root analytic conductor: \(3.60992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2663424,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−7.56738111269202357438702690211, −6.81173000092255075251437332159, −6.63257438834160544258861284685, −6.46249354410612141225423166895, −5.77904204714328784200418377249, −4.91997035368174263783342884746, −4.91585848792345990418783522770, −4.36159048011225245066599957337, −3.75890531283655162147766509504, −3.36779318987556938646346405293, −2.94610759542800375463162022317, −2.09209863939800055422786202913, −2.07543878259651085165424781495, −1.16529472267103972455357580300, 0, 1.16529472267103972455357580300, 2.07543878259651085165424781495, 2.09209863939800055422786202913, 2.94610759542800375463162022317, 3.36779318987556938646346405293, 3.75890531283655162147766509504, 4.36159048011225245066599957337, 4.91585848792345990418783522770, 4.91997035368174263783342884746, 5.77904204714328784200418377249, 6.46249354410612141225423166895, 6.63257438834160544258861284685, 6.81173000092255075251437332159, 7.56738111269202357438702690211

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