Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 17^{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  − 2·4-s − 8·7-s + 9-s + 4·16-s − 2·17-s + 18·23-s − 25-s + 16·28-s + 4·31-s − 2·36-s − 6·41-s − 12·47-s + 34·49-s − 8·63-s − 8·64-s + 4·68-s + 24·71-s + 4·73-s − 20·79-s + 81-s − 36·92-s − 32·97-s + 2·100-s + 10·103-s − 32·112-s − 18·113-s + 16·119-s + ⋯
L(s)  = 1  − 4-s − 3.02·7-s + 1/3·9-s + 16-s − 0.485·17-s + 3.75·23-s − 1/5·25-s + 3.02·28-s + 0.718·31-s − 1/3·36-s − 0.937·41-s − 1.75·47-s + 34/7·49-s − 1.00·63-s − 64-s + 0.485·68-s + 2.84·71-s + 0.468·73-s − 2.25·79-s + 1/9·81-s − 3.75·92-s − 3.24·97-s + 1/5·100-s + 0.985·103-s − 3.02·112-s − 1.69·113-s + 1.46·119-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  :  no
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$C_2$ \( 1 + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−9.250915914068804964862203626670, −8.618966621189813626434174913816, −8.228682176903090380178701350148, −7.31936717375093530826366945214, −6.80299500068960285512965295533, −6.66463575826783700250676263362, −6.19889841643513433803233134244, −5.21243235840224221221520663568, −5.14306131647545403565477997005, −4.18993695410695019989623309599, −3.63699123933177341557225053751, −3.00632648437536992753290178800, −2.85877051829134286082199945018, −1.08612370698107395863808052921, 0, 1.08612370698107395863808052921, 2.85877051829134286082199945018, 3.00632648437536992753290178800, 3.63699123933177341557225053751, 4.18993695410695019989623309599, 5.14306131647545403565477997005, 5.21243235840224221221520663568, 6.19889841643513433803233134244, 6.66463575826783700250676263362, 6.80299500068960285512965295533, 7.31936717375093530826366945214, 8.228682176903090380178701350148, 8.618966621189813626434174913816, 9.250915914068804964862203626670

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