Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{3} \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-s + 3-s − 4-s − 6-s + 3·8-s + 9-s + 8·11-s − 12-s − 4·13-s − 16-s − 18-s − 8·22-s + 3·24-s − 6·25-s + 4·26-s + 27-s − 5·32-s + 8·33-s − 36-s + 12·37-s − 4·39-s − 8·44-s − 48-s + 49-s + 6·50-s + 4·52-s − 54-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 2.41·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s − 0.235·18-s − 1.70·22-s + 0.612·24-s − 6/5·25-s + 0.784·26-s + 0.192·27-s − 0.883·32-s + 1.39·33-s − 1/6·36-s + 1.97·37-s − 0.640·39-s − 1.20·44-s − 0.144·48-s + 1/7·49-s + 0.848·50-s + 0.554·52-s − 0.136·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21168\)    =    \(2^{4} \cdot 3^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{21168} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 21168,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9399498356$
$L(\frac12)$  $\approx$  $0.9399498356$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−10.64068584777438676509120304615, −9.857567892373655758925106479846, −9.724661210726173973340366981146, −9.306486168015385400395632025578, −8.672365196683144779961098572501, −8.351153010775658504468507691128, −7.45452630662509700459332399892, −7.25047783802838427330028450541, −6.43539728173776997986575871832, −5.76945209288606009941542010236, −4.75834670654208540745088589071, −4.13559084050773741974089362056, −3.71279869000131289976682575519, −2.38203049551357769485459868076, −1.27003349896142462152543818564, 1.27003349896142462152543818564, 2.38203049551357769485459868076, 3.71279869000131289976682575519, 4.13559084050773741974089362056, 4.75834670654208540745088589071, 5.76945209288606009941542010236, 6.43539728173776997986575871832, 7.25047783802838427330028450541, 7.45452630662509700459332399892, 8.351153010775658504468507691128, 8.672365196683144779961098572501, 9.306486168015385400395632025578, 9.724661210726173973340366981146, 9.857567892373655758925106479846, 10.64068584777438676509120304615

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