Properties

Degree 4
Conductor $ 2^{5} \cdot 7^{2} \cdot 23^{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-s + 4-s + 2·7-s − 8-s − 2·9-s − 8·13-s − 2·14-s + 16-s + 2·18-s + 4·19-s − 10·25-s + 8·26-s + 2·28-s − 12·29-s − 32-s − 2·36-s − 4·38-s + 12·41-s + 16·43-s + 3·49-s + 10·50-s − 8·52-s − 2·56-s + 12·58-s − 4·63-s + 64-s − 8·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 2.21·13-s − 0.534·14-s + 1/4·16-s + 0.471·18-s + 0.917·19-s − 2·25-s + 1.56·26-s + 0.377·28-s − 2.22·29-s − 0.176·32-s − 1/3·36-s − 0.648·38-s + 1.87·41-s + 2.43·43-s + 3/7·49-s + 1.41·50-s − 1.10·52-s − 0.267·56-s + 1.57·58-s − 0.503·63-s + 1/8·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 829472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 829472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(829472\)    =    \(2^{5} \cdot 7^{2} \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{829472} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 829472,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.8185683604$
$L(\frac12)$  $\approx$  $0.8185683604$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;23\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
7$C_1$ \( ( 1 - T )^{2} \)
23$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.964944662527870856026164037982, −7.64178149897623079839720524725, −7.57571100088867902110310233811, −7.28614091898437125853646778423, −6.48583845948574201695144290108, −5.81994722151176032382577376975, −5.57928681742950427486583645839, −5.25222358335302626915983382813, −4.45632876697515521773485306871, −4.09738071226881210588087278636, −3.39198469441863826812749201545, −2.50137257751084499884008386915, −2.37043392826247813268923532045, −1.61447913787792106227712636889, −0.46915074201749145746453250386, 0.46915074201749145746453250386, 1.61447913787792106227712636889, 2.37043392826247813268923532045, 2.50137257751084499884008386915, 3.39198469441863826812749201545, 4.09738071226881210588087278636, 4.45632876697515521773485306871, 5.25222358335302626915983382813, 5.57928681742950427486583645839, 5.81994722151176032382577376975, 6.48583845948574201695144290108, 7.28614091898437125853646778423, 7.57571100088867902110310233811, 7.64178149897623079839720524725, 7.964944662527870856026164037982

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