Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{3} \cdot 19^{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  + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 12-s + 12·13-s − 16-s + 18-s + 8·23-s − 3·24-s − 6·25-s + 12·26-s + 27-s + 5·32-s − 36-s − 20·37-s + 12·39-s + 8·46-s + 24·47-s − 48-s − 14·49-s − 6·50-s − 12·52-s + 54-s − 24·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 3.32·13-s − 1/4·16-s + 0.235·18-s + 1.66·23-s − 0.612·24-s − 6/5·25-s + 2.35·26-s + 0.192·27-s + 0.883·32-s − 1/6·36-s − 3.28·37-s + 1.92·39-s + 1.17·46-s + 3.50·47-s − 0.144·48-s − 2·49-s − 0.848·50-s − 1.66·52-s + 0.136·54-s − 3.12·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(155952\)    =    \(2^{4} \cdot 3^{3} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{155952} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 155952,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.720199700$
$L(\frac12)$  $\approx$  $2.720199700$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.092693734190788229037074202448, −9.032116632714081125535639739797, −8.263513432862467471567669978548, −8.062008804214737453511932083588, −7.33667434445098454643296446245, −6.46787589567348650226556085189, −6.39584814801769229586264060019, −5.69178884171323828176864961556, −5.22523151161492305197654101068, −4.56991321080597356703354311172, −3.84859941092912106278451327058, −3.44365518362789223021999993571, −3.25148783575260438600086430315, −1.95255670636420425042320954209, −1.07628396925620245570845051253, 1.07628396925620245570845051253, 1.95255670636420425042320954209, 3.25148783575260438600086430315, 3.44365518362789223021999993571, 3.84859941092912106278451327058, 4.56991321080597356703354311172, 5.22523151161492305197654101068, 5.69178884171323828176864961556, 6.39584814801769229586264060019, 6.46787589567348650226556085189, 7.33667434445098454643296446245, 8.062008804214737453511932083588, 8.263513432862467471567669978548, 9.032116632714081125535639739797, 9.092693734190788229037074202448

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