Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{4} \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  − 4·7-s + 8·19-s + 10·25-s − 12·29-s + 8·31-s + 4·37-s + 9·49-s − 12·53-s + 24·59-s + 24·83-s + 8·103-s + 20·109-s + 12·113-s + 10·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s − 40·175-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.83·19-s + 2·25-s − 2.22·29-s + 1.43·31-s + 0.657·37-s + 9/7·49-s − 1.64·53-s + 3.12·59-s + 2.63·83-s + 0.788·103-s + 1.91·109-s + 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-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 + 2·169-s + 0.0760·173-s − 3.02·175-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16257024\)    =    \(2^{12} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4032} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 16257024,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.588544701$
$L(\frac12)$  $\approx$  $2.588544701$
$L(\frac{3}{2})$   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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−8.729042830694378557166007395628, −8.248255019414782111912352080542, −7.73354841966454197465585841455, −7.60421089041191712934912239510, −7.09079297781628177176323454343, −6.79381908308950977354078845526, −6.43564900400331234984549772955, −6.18747274383979713621610425701, −5.52796586846360439417760801881, −5.41854346842618749203109540012, −4.92799333860470475965942300229, −4.50334866010050679100304505693, −3.92320028349439863829764838679, −3.49587098587077795521985227194, −3.20728904815708636043613575844, −2.86827251153802823377210104329, −2.33926009880233913779483369028, −1.72722559933542220662194048639, −0.850528696633354334791286061642, −0.62770340494346190445203198598, 0.62770340494346190445203198598, 0.850528696633354334791286061642, 1.72722559933542220662194048639, 2.33926009880233913779483369028, 2.86827251153802823377210104329, 3.20728904815708636043613575844, 3.49587098587077795521985227194, 3.92320028349439863829764838679, 4.50334866010050679100304505693, 4.92799333860470475965942300229, 5.41854346842618749203109540012, 5.52796586846360439417760801881, 6.18747274383979713621610425701, 6.43564900400331234984549772955, 6.79381908308950977354078845526, 7.09079297781628177176323454343, 7.60421089041191712934912239510, 7.73354841966454197465585841455, 8.248255019414782111912352080542, 8.729042830694378557166007395628

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