Properties

Degree 4
Conductor $ 2^{7} \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  + 2·5-s − 8·7-s − 6·9-s + 4·11-s + 8·19-s + 3·25-s − 16·35-s + 12·37-s − 16·43-s − 12·45-s + 34·49-s + 12·53-s + 8·55-s + 48·63-s − 32·77-s + 27·81-s − 32·83-s − 12·89-s + 16·95-s − 28·97-s − 24·99-s + 36·113-s + 5·121-s + 4·125-s + 127-s + 131-s − 64·133-s + ⋯
L(s)  = 1  + 0.894·5-s − 3.02·7-s − 2·9-s + 1.20·11-s + 1.83·19-s + 3/5·25-s − 2.70·35-s + 1.97·37-s − 2.43·43-s − 1.78·45-s + 34/7·49-s + 1.64·53-s + 1.07·55-s + 6.04·63-s − 3.64·77-s + 3·81-s − 3.51·83-s − 1.27·89-s + 1.64·95-s − 2.84·97-s − 2.41·99-s + 3.38·113-s + 5/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(387200\)    =    \(2^{7} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{387200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 387200,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.563879011661793903559115865259, −8.223522789927760706880481247520, −7.17333521505739308879256846630, −6.97238544391138654785050907556, −6.51302590756949038823070214927, −6.02838569034650388568211971598, −5.70093888866808737514223874840, −5.54153279390545819436346988340, −4.53243901735140236687311130642, −3.66574171865656283286960534597, −3.26180587408034592610487247010, −2.94467874037426815586288045320, −2.44821359744315819514002541053, −1.09598094259623312561400782139, 0, 1.09598094259623312561400782139, 2.44821359744315819514002541053, 2.94467874037426815586288045320, 3.26180587408034592610487247010, 3.66574171865656283286960534597, 4.53243901735140236687311130642, 5.54153279390545819436346988340, 5.70093888866808737514223874840, 6.02838569034650388568211971598, 6.51302590756949038823070214927, 6.97238544391138654785050907556, 7.17333521505739308879256846630, 8.223522789927760706880481247520, 8.563879011661793903559115865259

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