Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 7^{3} $
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  + 4·5-s − 7-s − 3·9-s − 12·17-s + 2·25-s − 4·35-s − 4·37-s + 4·41-s − 8·43-s − 12·45-s − 16·47-s + 49-s + 3·63-s − 8·67-s + 32·79-s + 9·81-s + 16·83-s − 48·85-s − 12·89-s + 4·101-s − 20·109-s + 12·119-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s − 9-s − 2.91·17-s + 2/5·25-s − 0.676·35-s − 0.657·37-s + 0.624·41-s − 1.21·43-s − 1.78·45-s − 2.33·47-s + 1/7·49-s + 0.377·63-s − 0.977·67-s + 3.60·79-s + 81-s + 1.75·83-s − 5.20·85-s − 1.27·89-s + 0.398·101-s − 1.91·109-s + 1.10·119-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(197568\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{197568} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 197568,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−9.096051595987974270124894157352, −8.373309408931680629637484917349, −8.188420417684287628984093773695, −7.32119224555753352111842303305, −6.52917361486339326450126089747, −6.42262084813563967264199691637, −6.17908275353188198229198176353, −5.22316409435338364257345412913, −5.16177791023533130836977920691, −4.33405029272495229169198202320, −3.58097051005902557419651238987, −2.79183800612725741835263061431, −2.16319834122457639545883386464, −1.80766814068560455754339816128, 0, 1.80766814068560455754339816128, 2.16319834122457639545883386464, 2.79183800612725741835263061431, 3.58097051005902557419651238987, 4.33405029272495229169198202320, 5.16177791023533130836977920691, 5.22316409435338364257345412913, 6.17908275353188198229198176353, 6.42262084813563967264199691637, 6.52917361486339326450126089747, 7.32119224555753352111842303305, 8.188420417684287628984093773695, 8.373309408931680629637484917349, 9.096051595987974270124894157352

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