Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{2} \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 + 4-s − 8·7-s − 8-s + 9-s + 8·14-s + 16-s + 12·17-s − 18-s + 25-s − 8·28-s + 16·31-s − 32-s − 12·34-s + 36-s − 12·41-s + 34·49-s − 50-s + 8·56-s − 16·62-s − 8·63-s + 64-s + 12·68-s − 72-s + 4·73-s + 16·79-s + 81-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 3.02·7-s − 0.353·8-s + 1/3·9-s + 2.13·14-s + 1/4·16-s + 2.91·17-s − 0.235·18-s + 1/5·25-s − 1.51·28-s + 2.87·31-s − 0.176·32-s − 2.05·34-s + 1/6·36-s − 1.87·41-s + 34/7·49-s − 0.141·50-s + 1.06·56-s − 2.03·62-s − 1.00·63-s + 1/8·64-s + 1.45·68-s − 0.117·72-s + 0.468·73-s + 1.80·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(28800\)    =    \(2^{7} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{28800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 28800,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.6620615532$
$L(\frac12)$  $\approx$  $0.6620615532$
$L(\frac{3}{2})$   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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.39160509435207624542212960871, −9.861778856867545654535992942161, −9.651683174776301621329437840888, −9.305587122869086271308905422277, −8.381333480789723559704518400634, −7.941231322257043910223245152113, −7.26843774815325716692994821945, −6.55369649177171482035005226183, −6.42217617652666865799421983967, −5.77106186730486543398971536544, −4.95941068749140901060295823906, −3.58551752676900090223959184299, −3.36585804145949552210873743484, −2.69384090145739289224125918540, −0.889333176383366415114391202530, 0.889333176383366415114391202530, 2.69384090145739289224125918540, 3.36585804145949552210873743484, 3.58551752676900090223959184299, 4.95941068749140901060295823906, 5.77106186730486543398971536544, 6.42217617652666865799421983967, 6.55369649177171482035005226183, 7.26843774815325716692994821945, 7.941231322257043910223245152113, 8.381333480789723559704518400634, 9.305587122869086271308905422277, 9.651683174776301621329437840888, 9.861778856867545654535992942161, 10.39160509435207624542212960871

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