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

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 4·19-s + 4·25-s + 12·29-s + 16·31-s − 8·37-s + 24·47-s − 3·49-s − 12·53-s + 24·59-s − 32·103-s − 4·109-s − 12·113-s + 16·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 8·175-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.917·19-s + 4/5·25-s + 2.22·29-s + 2.87·31-s − 1.31·37-s + 3.50·47-s − 3/7·49-s − 1.64·53-s + 3.12·59-s − 3.15·103-s − 0.383·109-s − 1.12·113-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-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 + 2/13·169-s + 0.0760·173-s − 0.604·175-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  =  $(4,\ 16257024,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.132667910$
$L(\frac12)$  $\approx$  $3.132667910$
$L(\frac{3}{2})$   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$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 170 T^{2} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.503764164737708725993132932199, −8.469204947544295420021501899176, −7.900989008141513844519779175484, −7.53688459541587718216138216824, −7.04553103045493731933468230575, −6.79004559055561789030135910470, −6.37479369745579898372395713839, −6.28730855364845039368042757494, −5.63001794155721557041355695650, −5.10885848877651099318950978038, −5.08551732690939680305511450819, −4.40567675297270199358108154002, −4.05633313539108606858140977856, −3.68441074831981262996201358204, −2.95363662569522248918609961315, −2.71239194733128106647376662670, −2.59502862412262800116140544329, −1.59666040841639956781098124158, −0.947852183210974502938628026102, −0.65312076414784849846568338763, 0.65312076414784849846568338763, 0.947852183210974502938628026102, 1.59666040841639956781098124158, 2.59502862412262800116140544329, 2.71239194733128106647376662670, 2.95363662569522248918609961315, 3.68441074831981262996201358204, 4.05633313539108606858140977856, 4.40567675297270199358108154002, 5.08551732690939680305511450819, 5.10885848877651099318950978038, 5.63001794155721557041355695650, 6.28730855364845039368042757494, 6.37479369745579898372395713839, 6.79004559055561789030135910470, 7.04553103045493731933468230575, 7.53688459541587718216138216824, 7.900989008141513844519779175484, 8.469204947544295420021501899176, 8.503764164737708725993132932199

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