Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 17^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s + 9-s − 16-s − 2·17-s + 18-s − 10·25-s + 5·32-s − 2·34-s − 36-s − 16·47-s − 2·49-s − 10·50-s + 7·64-s + 2·68-s − 3·72-s + 81-s − 20·89-s − 16·94-s − 2·98-s + 10·100-s − 6·121-s + 127-s − 3·128-s + 131-s + 6·136-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1/3·9-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 2·25-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 2.33·47-s − 2/7·49-s − 1.41·50-s + 7/8·64-s + 0.242·68-s − 0.353·72-s + 1/9·81-s − 2.11·89-s − 1.65·94-s − 0.202·98-s + 100-s − 0.545·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.514·136-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(166464\)    =    \(2^{6} \cdot 3^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{166464} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 166464,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−8.990627621060578483098017594856, −8.466527665107842666653023212386, −8.040243271667994896507365067817, −7.61015041301891599610023453537, −6.85505714070999859977992001649, −6.45990116372875969103688741620, −5.87647456361852564136261392180, −5.45184408676781858561883617452, −4.81099171488550293269767138312, −4.36159397767822262013932326900, −3.79884678729776031688190779911, −3.28532150713041810310982697520, −2.45336032252096177013300497760, −1.56769530060653144854680691564, 0, 1.56769530060653144854680691564, 2.45336032252096177013300497760, 3.28532150713041810310982697520, 3.79884678729776031688190779911, 4.36159397767822262013932326900, 4.81099171488550293269767138312, 5.45184408676781858561883617452, 5.87647456361852564136261392180, 6.45990116372875969103688741620, 6.85505714070999859977992001649, 7.61015041301891599610023453537, 8.040243271667994896507365067817, 8.466527665107842666653023212386, 8.990627621060578483098017594856

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