Properties

Label 4-1632e2-1.1-c1e2-0-51
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 8·11-s + 2·17-s − 8·19-s − 6·25-s − 4·27-s + 16·33-s − 12·41-s − 8·43-s + 2·49-s − 4·51-s + 16·57-s + 8·59-s + 24·67-s + 20·73-s + 12·75-s + 5·81-s − 8·83-s − 12·89-s − 12·97-s − 24·99-s − 40·107-s − 12·113-s + 26·121-s + 24·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 2.41·11-s + 0.485·17-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 + 2/7·49-s − 0.560·51-s + 2.11·57-s + 1.04·59-s + 2.93·67-s + 2.34·73-s + 1.38·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s − 1.21·97-s − 2.41·99-s − 3.86·107-s − 1.12·113-s + 2.36·121-s + 2.16·123-s + 0.0887·127-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: \(2\)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−6.98083747403506855455974764057, −6.62202948008203069684059614968, −6.59291711794167281790003444711, −5.67636732029647007076099939422, −5.51324452492528964790663180164, −5.13001221098195863338015289116, −4.91029437030238646203911839660, −4.02556354189723006943550620116, −3.92247845698833805317728267090, −3.12397148446932752142076853180, −2.35058397621900011908829435300, −2.18463537719495235457264747710, −1.26926232847077019067068429617, 0, 0, 1.26926232847077019067068429617, 2.18463537719495235457264747710, 2.35058397621900011908829435300, 3.12397148446932752142076853180, 3.92247845698833805317728267090, 4.02556354189723006943550620116, 4.91029437030238646203911839660, 5.13001221098195863338015289116, 5.51324452492528964790663180164, 5.67636732029647007076099939422, 6.59291711794167281790003444711, 6.62202948008203069684059614968, 6.98083747403506855455974764057

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