Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{6} \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  − 7-s + 16·19-s + 7·25-s + 12·29-s − 10·31-s + 8·37-s − 12·47-s − 6·49-s + 6·53-s − 24·59-s − 30·83-s + 32·103-s + 40·109-s + 24·113-s − 5·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  − 0.377·7-s + 3.67·19-s + 7/5·25-s + 2.22·29-s − 1.79·31-s + 1.31·37-s − 1.75·47-s − 6/7·49-s + 0.824·53-s − 3.12·59-s − 3.29·83-s + 3.15·103-s + 3.83·109-s + 2.25·113-s − 0.454·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.884832359\)
\(L(\frac12)\)  \(\approx\)  \(2.884832359\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
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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

−9.154480413383507872776540143350, −8.485748344008287363452887878158, −8.230659355500302233329943368466, −7.56147787022240455212107554339, −7.44158493876838305652500989366, −7.22293875315507179668820537662, −6.64736846153915600103018875532, −6.23791272768971002080021076306, −5.90972994336019980328210711859, −5.43383286617479293384194166205, −5.05005180770035736117198639534, −4.62065043187181382590848178771, −4.47239690492159970291767306484, −3.41458956676008800958214186861, −3.26290870978848290754273136398, −3.09037940669844404205461566467, −2.50456312538850418832409855782, −1.58946466492516623393880001500, −1.20458629331792589701431308306, −0.60253261047836479686556714910, 0.60253261047836479686556714910, 1.20458629331792589701431308306, 1.58946466492516623393880001500, 2.50456312538850418832409855782, 3.09037940669844404205461566467, 3.26290870978848290754273136398, 3.41458956676008800958214186861, 4.47239690492159970291767306484, 4.62065043187181382590848178771, 5.05005180770035736117198639534, 5.43383286617479293384194166205, 5.90972994336019980328210711859, 6.23791272768971002080021076306, 6.64736846153915600103018875532, 7.22293875315507179668820537662, 7.44158493876838305652500989366, 7.56147787022240455212107554339, 8.230659355500302233329943368466, 8.485748344008287363452887878158, 9.154480413383507872776540143350

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