Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{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 − 3·8-s − 3·9-s − 2·11-s + 4·13-s − 16-s − 3·18-s − 2·22-s + 8·23-s + 25-s + 4·26-s + 5·32-s + 3·36-s − 4·37-s + 2·44-s + 8·46-s − 24·47-s − 14·49-s + 50-s − 4·52-s + 8·59-s − 20·61-s + 7·64-s + 16·71-s + 9·72-s + 28·73-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s − 0.603·11-s + 1.10·13-s − 1/4·16-s − 0.707·18-s − 0.426·22-s + 1.66·23-s + 1/5·25-s + 0.784·26-s + 0.883·32-s + 1/2·36-s − 0.657·37-s + 0.301·44-s + 1.17·46-s − 3.50·47-s − 2·49-s + 0.141·50-s − 0.554·52-s + 1.04·59-s − 2.56·61-s + 7/8·64-s + 1.89·71-s + 1.06·72-s + 3.27·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(435600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{435600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 435600,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;11\}$, 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$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−8.248642768475844318802620430882, −8.227294201073412154999968421070, −7.52567387860361416690294259657, −6.73114079621071100714169312036, −6.32256355952646480381816699905, −6.16821463973303244938343944630, −5.19072755325840146238482378089, −5.09766059609940450216711100862, −4.80763834609260757451310676100, −3.82573290551710488813917015410, −3.25423179226253980082861279896, −3.19776287627196162761105647639, −2.27052010539287522961646416611, −1.21814035480499692293997625850, 0, 1.21814035480499692293997625850, 2.27052010539287522961646416611, 3.19776287627196162761105647639, 3.25423179226253980082861279896, 3.82573290551710488813917015410, 4.80763834609260757451310676100, 5.09766059609940450216711100862, 5.19072755325840146238482378089, 6.16821463973303244938343944630, 6.32256355952646480381816699905, 6.73114079621071100714169312036, 7.52567387860361416690294259657, 8.227294201073412154999968421070, 8.248642768475844318802620430882

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