Properties

Degree 4
Conductor $ 2^{5} \cdot 3^{3} \cdot 5^{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 − 8-s + 9-s + 12-s + 4·13-s + 16-s − 18-s − 24-s + 25-s − 4·26-s + 27-s − 32-s + 36-s + 4·37-s + 4·39-s + 48-s + 2·49-s − 50-s + 4·52-s − 54-s − 20·61-s + 64-s − 72-s + 4·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.235·18-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.176·32-s + 1/6·36-s + 0.657·37-s + 0.640·39-s + 0.144·48-s + 2/7·49-s − 0.141·50-s + 0.554·52-s − 0.136·54-s − 2.56·61-s + 1/8·64-s − 0.117·72-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21600\)    =    \(2^{5} \cdot 3^{3} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{21600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 21600,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.081141989$
$L(\frac12)$  $\approx$  $1.081141989$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.67015888052155601125139797767, −10.39160509435207624542212960871, −9.415401762585890013045544643092, −9.305587122869086271308905422277, −8.713350404322888410914105544855, −7.941231322257043910223245152113, −7.88297907167905508023143292047, −6.96170559755127999142057961261, −6.42217617652666865799421983967, −5.85236328773314391596499497567, −4.97026786036012165020386374880, −4.08471431361169401081275954745, −3.36585804145949552210873743484, −2.50374962358163435732612030839, −1.36243712364701817507861978801, 1.36243712364701817507861978801, 2.50374962358163435732612030839, 3.36585804145949552210873743484, 4.08471431361169401081275954745, 4.97026786036012165020386374880, 5.85236328773314391596499497567, 6.42217617652666865799421983967, 6.96170559755127999142057961261, 7.88297907167905508023143292047, 7.941231322257043910223245152113, 8.713350404322888410914105544855, 9.305587122869086271308905422277, 9.415401762585890013045544643092, 10.39160509435207624542212960871, 10.67015888052155601125139797767

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