Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 4·5-s − 3·8-s + 9-s − 4·10-s + 12·13-s − 16-s − 12·17-s + 18-s + 4·20-s + 2·25-s + 12·26-s + 4·29-s + 5·32-s − 12·34-s − 36-s − 20·37-s + 12·40-s − 4·41-s − 4·45-s − 14·49-s + 2·50-s − 12·52-s − 12·53-s + 4·58-s − 4·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s + 1/3·9-s − 1.26·10-s + 3.32·13-s − 1/4·16-s − 2.91·17-s + 0.235·18-s + 0.894·20-s + 2/5·25-s + 2.35·26-s + 0.742·29-s + 0.883·32-s − 2.05·34-s − 1/6·36-s − 3.28·37-s + 1.89·40-s − 0.624·41-s − 0.596·45-s − 2·49-s + 0.282·50-s − 1.66·52-s − 1.64·53-s + 0.525·58-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(51984\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{51984} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 51984,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−9.717867966311693207189859225084, −9.032116632714081125535639739797, −8.673289841767403383115719575464, −8.263513432862467471567669978548, −8.099644677533565214444467559710, −6.98236656892345640095319736227, −6.46787589567348650226556085189, −6.30774150521524867405980638808, −5.28029214515602867786662657712, −4.56991321080597356703354311172, −4.16838172205224943400021226417, −3.49609663069962101813276746331, −3.44365518362789223021999993571, −1.72203721066548968493435337820, 0, 1.72203721066548968493435337820, 3.44365518362789223021999993571, 3.49609663069962101813276746331, 4.16838172205224943400021226417, 4.56991321080597356703354311172, 5.28029214515602867786662657712, 6.30774150521524867405980638808, 6.46787589567348650226556085189, 6.98236656892345640095319736227, 8.099644677533565214444467559710, 8.263513432862467471567669978548, 8.673289841767403383115719575464, 9.032116632714081125535639739797, 9.717867966311693207189859225084

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