Properties

Degree 4
Conductor $ 2^{7} \cdot 5^{2} \cdot 19^{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 − 6·9-s + 4·17-s + 4·19-s + 3·25-s − 16·31-s − 12·45-s + 2·49-s − 8·59-s − 4·61-s + 16·67-s − 12·73-s + 27·81-s + 8·85-s + 8·95-s + 12·101-s + 8·103-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s − 32·155-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s + 0.970·17-s + 0.917·19-s + 3/5·25-s − 2.87·31-s − 1.78·45-s + 2/7·49-s − 1.04·59-s − 0.512·61-s + 1.95·67-s − 1.40·73-s + 3·81-s + 0.867·85-s + 0.820·95-s + 1.19·101-s + 0.788·103-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s − 2.57·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1155200\)    =    \(2^{7} \cdot 5^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1155200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 1155200,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−7.84210102043042646384025401352, −7.46212672600372947009312463710, −6.97238544391138654785050907556, −6.37399344243820867996093621951, −5.86830391534173005162345630789, −5.70093888866808737514223874840, −5.20324548069731475827972490709, −5.02429913120480640214209205427, −4.05130010083233398855700190379, −3.32072280545665331707726425753, −3.26180587408034592610487247010, −2.47692318206012541987277725331, −1.96992891210281049758559614540, −1.13847461285225699369638878979, 0, 1.13847461285225699369638878979, 1.96992891210281049758559614540, 2.47692318206012541987277725331, 3.26180587408034592610487247010, 3.32072280545665331707726425753, 4.05130010083233398855700190379, 5.02429913120480640214209205427, 5.20324548069731475827972490709, 5.70093888866808737514223874840, 5.86830391534173005162345630789, 6.37399344243820867996093621951, 6.97238544391138654785050907556, 7.46212672600372947009312463710, 7.84210102043042646384025401352

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