Properties

Degree 4
Conductor $ 2^{5} \cdot 3^{3} \cdot 11^{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-s + 3-s + 4-s − 6-s − 8-s + 9-s − 2·11-s + 12-s − 8·13-s + 16-s − 18-s + 2·22-s + 12·23-s − 24-s − 10·25-s + 8·26-s + 27-s − 32-s − 2·33-s + 36-s − 20·37-s − 8·39-s − 2·44-s − 12·46-s − 12·47-s + 48-s − 10·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 2.21·13-s + 1/4·16-s − 0.235·18-s + 0.426·22-s + 2.50·23-s − 0.204·24-s − 2·25-s + 1.56·26-s + 0.192·27-s − 0.176·32-s − 0.348·33-s + 1/6·36-s − 3.28·37-s − 1.28·39-s − 0.301·44-s − 1.76·46-s − 1.75·47-s + 0.144·48-s − 1.42·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(104544\)    =    \(2^{5} \cdot 3^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{104544} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 104544,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.245228842316309160133095693960, −8.958855021658912251315559727512, −8.129852226719742477393784673308, −8.013104010237496671597514230187, −7.33998580522115612678977642043, −6.79963832425739493066244038791, −6.77506565559004391547494912024, −5.43952609375822489573657963774, −5.21184114914749280771382385734, −4.69333471332706230755932945396, −3.64282367595579751815074401037, −3.08131485219503756183708188012, −2.37545091357255938865951638025, −1.70794665150414314375514308074, 0, 1.70794665150414314375514308074, 2.37545091357255938865951638025, 3.08131485219503756183708188012, 3.64282367595579751815074401037, 4.69333471332706230755932945396, 5.21184114914749280771382385734, 5.43952609375822489573657963774, 6.77506565559004391547494912024, 6.79963832425739493066244038791, 7.33998580522115612678977642043, 8.013104010237496671597514230187, 8.129852226719742477393784673308, 8.958855021658912251315559727512, 9.245228842316309160133095693960

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