Properties

Degree 4
Conductor $ 2^{7} \cdot 5^{2} \cdot 13^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 6·3-s + 4-s − 5-s − 6·6-s + 8-s + 21·9-s − 10-s − 6·12-s − 2·13-s + 6·15-s + 16-s + 21·18-s − 20-s − 6·24-s − 4·25-s − 2·26-s − 54·27-s + 6·30-s + 8·31-s + 32-s + 21·36-s + 6·37-s + 12·39-s − 40-s − 10·43-s − 21·45-s + ⋯
L(s)  = 1  + 0.707·2-s − 3.46·3-s + 1/2·4-s − 0.447·5-s − 2.44·6-s + 0.353·8-s + 7·9-s − 0.316·10-s − 1.73·12-s − 0.554·13-s + 1.54·15-s + 1/4·16-s + 4.94·18-s − 0.223·20-s − 1.22·24-s − 4/5·25-s − 0.392·26-s − 10.3·27-s + 1.09·30-s + 1.43·31-s + 0.176·32-s + 7/2·36-s + 0.986·37-s + 1.92·39-s − 0.158·40-s − 1.52·43-s − 3.13·45-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(540800\)    =    \(2^{7} \cdot 5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{540800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 540800,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 - T \)
5$C_2$ \( 1 + T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
show more
show less
\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.947525996725592169519910140162, −7.45985560667588965978165906144, −7.06351514889423061352505396698, −6.57298406981565936099456689488, −6.32528200643262897349180267262, −5.88908450915949561478136129218, −5.41696520049424981149332978983, −5.09892068910016055764395140389, −4.61153453491182141362175201622, −4.28784246485114607808557377793, −3.74839125626037417787439212589, −2.72660213623419522592921634901, −1.67224901915905998978362533056, −0.881323602688431658600765351508, 0, 0.881323602688431658600765351508, 1.67224901915905998978362533056, 2.72660213623419522592921634901, 3.74839125626037417787439212589, 4.28784246485114607808557377793, 4.61153453491182141362175201622, 5.09892068910016055764395140389, 5.41696520049424981149332978983, 5.88908450915949561478136129218, 6.32528200643262897349180267262, 6.57298406981565936099456689488, 7.06351514889423061352505396698, 7.45985560667588965978165906144, 7.947525996725592169519910140162

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