Properties

Degree 4
Conductor $ 2^{6} \cdot 5^{2} \cdot 7^{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 − 8·11-s + 16·19-s − 25-s + 12·29-s + 16·31-s + 4·41-s − 12·45-s + 49-s − 16·55-s − 12·61-s − 16·71-s + 32·79-s + 27·81-s − 12·89-s + 32·95-s + 48·99-s + 4·101-s − 20·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s − 2.41·11-s + 3.67·19-s − 1/5·25-s + 2.22·29-s + 2.87·31-s + 0.624·41-s − 1.78·45-s + 1/7·49-s − 2.15·55-s − 1.53·61-s − 1.89·71-s + 3.60·79-s + 3·81-s − 1.27·89-s + 3.28·95-s + 4.82·99-s + 0.398·101-s − 1.91·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{78400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 78400,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.368178000$
$L(\frac12)$  $\approx$  $1.368178000$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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

−9.694440338392361408863185872440, −9.457349313423188294666539437989, −8.660773281175179322159496718137, −8.188420417684287628984093773695, −7.81362030580797675551267597140, −7.47489161542833731239859573103, −6.42262084813563967264199691637, −6.06292563014554710285489032478, −5.38565578310165680979646788554, −5.22316409435338364257345412913, −4.71019542661529810418780789156, −3.18975677352219866037498641634, −2.79183800612725741835263061431, −2.62755636242588070812542768623, −0.916171764469637391994006820348, 0.916171764469637391994006820348, 2.62755636242588070812542768623, 2.79183800612725741835263061431, 3.18975677352219866037498641634, 4.71019542661529810418780789156, 5.22316409435338364257345412913, 5.38565578310165680979646788554, 6.06292563014554710285489032478, 6.42262084813563967264199691637, 7.47489161542833731239859573103, 7.81362030580797675551267597140, 8.188420417684287628984093773695, 8.660773281175179322159496718137, 9.457349313423188294666539437989, 9.694440338392361408863185872440

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