Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \cdot 5^{4} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·17-s + 2·19-s + 2·23-s + 6·32-s − 4·34-s + 4·38-s + 4·46-s − 2·47-s + 2·61-s + 7·64-s − 6·68-s + 6·76-s − 4·83-s + 6·92-s − 4·94-s − 4·107-s − 2·109-s + 2·113-s + 4·122-s + 127-s + 8·128-s + 131-s − 8·136-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·17-s + 2·19-s + 2·23-s + 6·32-s − 4·34-s + 4·38-s + 4·46-s − 2·47-s + 2·61-s + 7·64-s − 6·68-s + 6·76-s − 4·83-s + 6·92-s − 4·94-s − 4·107-s − 2·109-s + 2·113-s + 4·122-s + 127-s + 8·128-s + 131-s − 8·136-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{3600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 12960000,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $6.552958166$
$L(\frac12)$  $\approx$  $6.552958166$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−8.790992238917499576246919786578, −8.478989919133056169283558066796, −8.026537898858517693773409798944, −7.57513745641619228211733496290, −7.18856308503068707964419483434, −6.93020166853298988510859527003, −6.55904275422440354084720007923, −6.47241231574239751317512153207, −5.63142473548678773609038152371, −5.50470912544726460002159708056, −4.97765505645303978347063780143, −4.94797969430622192602973608200, −4.20836516324813963644486868443, −4.11156005976907006916252028381, −3.40537873563978453697267780075, −3.13808068165915260415013603914, −2.58104332158666329411768478999, −2.43928531236859538920031795290, −1.44904393535381829465750433600, −1.28524364473144289524865885834, 1.28524364473144289524865885834, 1.44904393535381829465750433600, 2.43928531236859538920031795290, 2.58104332158666329411768478999, 3.13808068165915260415013603914, 3.40537873563978453697267780075, 4.11156005976907006916252028381, 4.20836516324813963644486868443, 4.94797969430622192602973608200, 4.97765505645303978347063780143, 5.50470912544726460002159708056, 5.63142473548678773609038152371, 6.47241231574239751317512153207, 6.55904275422440354084720007923, 6.93020166853298988510859527003, 7.18856308503068707964419483434, 7.57513745641619228211733496290, 8.026537898858517693773409798944, 8.478989919133056169283558066796, 8.790992238917499576246919786578

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