Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{3} \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  − 2·2-s − 3-s + 2·4-s + 2·6-s + 9-s + 2·11-s − 2·12-s + 4·13-s − 4·16-s − 2·18-s − 4·22-s − 8·23-s − 25-s − 8·26-s − 27-s + 8·32-s − 2·33-s + 2·36-s − 4·39-s + 4·44-s + 16·46-s − 18·47-s + 4·48-s + 11·49-s + 2·50-s + 8·52-s + 2·54-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 1.10·13-s − 16-s − 0.471·18-s − 0.852·22-s − 1.66·23-s − 1/5·25-s − 1.56·26-s − 0.192·27-s + 1.41·32-s − 0.348·33-s + 1/3·36-s − 0.640·39-s + 0.603·44-s + 2.35·46-s − 2.62·47-s + 0.577·48-s + 11/7·49-s + 0.282·50-s + 1.10·52-s + 0.272·54-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  =  \(1\)
Selberg data  =  \((4,\ 155952,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( 1 + T \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−8.836427381447004052087018681826, −8.742321760001961651389979974377, −8.160007849857414045140648086964, −7.47848778675296589128891353641, −7.43632371515983923998772054443, −6.41173993136661837405396202601, −6.29636417830616150950587212845, −5.80065514036351948381475791738, −4.91760568249510581017872290537, −4.32680757840779686690441622638, −3.81591641511052084615077615666, −2.94683125587980903660881898164, −1.81200784323266148009001357293, −1.34947676285250801422046885118, 0, 1.34947676285250801422046885118, 1.81200784323266148009001357293, 2.94683125587980903660881898164, 3.81591641511052084615077615666, 4.32680757840779686690441622638, 4.91760568249510581017872290537, 5.80065514036351948381475791738, 6.29636417830616150950587212845, 6.41173993136661837405396202601, 7.43632371515983923998772054443, 7.47848778675296589128891353641, 8.160007849857414045140648086964, 8.742321760001961651389979974377, 8.836427381447004052087018681826

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