Properties

Degree 4
Conductor $ 2^{6} \cdot 5^{2} \cdot 11^{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  + 2·5-s − 6·9-s + 4·11-s + 8·23-s + 3·25-s − 16·31-s + 12·37-s − 12·45-s + 8·47-s + 2·49-s + 12·53-s + 8·55-s − 8·59-s + 16·67-s + 27·81-s − 12·89-s − 28·97-s − 24·99-s + 8·103-s + 36·113-s + 16·115-s + 5·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s + 1.20·11-s + 1.66·23-s + 3/5·25-s − 2.87·31-s + 1.97·37-s − 1.78·45-s + 1.16·47-s + 2/7·49-s + 1.64·53-s + 1.07·55-s − 1.04·59-s + 1.95·67-s + 3·81-s − 1.27·89-s − 2.84·97-s − 2.41·99-s + 0.788·103-s + 3.38·113-s + 1.49·115-s + 5/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{193600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 193600,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.807192052$
$L(\frac12)$  $\approx$  $1.807192052$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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

−9.022035638415746178025932660186, −8.903529359998922904579331677853, −8.319898091581096916744910484661, −7.70303958223137845782045497044, −6.97238544391138654785050907556, −6.81644823665973489085396145114, −5.94028444997569064994570219738, −5.70093888866808737514223874840, −5.41810682445711458041939493554, −4.62487051901560546933133606596, −3.87138095586879394292498913230, −3.26180587408034592610487247010, −2.64264527642459335277526208075, −1.99873426678938511502281122159, −0.880315059374556524645724595724, 0.880315059374556524645724595724, 1.99873426678938511502281122159, 2.64264527642459335277526208075, 3.26180587408034592610487247010, 3.87138095586879394292498913230, 4.62487051901560546933133606596, 5.41810682445711458041939493554, 5.70093888866808737514223874840, 5.94028444997569064994570219738, 6.81644823665973489085396145114, 6.97238544391138654785050907556, 7.70303958223137845782045497044, 8.319898091581096916744910484661, 8.903529359998922904579331677853, 9.022035638415746178025932660186

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