Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{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  − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 9-s − 2·11-s − 4·14-s + 5·16-s − 2·18-s + 4·22-s + 12·23-s − 10·25-s + 6·28-s + 12·29-s − 6·32-s + 3·36-s − 20·37-s + 16·43-s − 6·44-s − 24·46-s − 3·49-s + 20·50-s − 8·56-s − 24·58-s + 2·63-s + 7·64-s − 8·67-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 1/3·9-s − 0.603·11-s − 1.06·14-s + 5/4·16-s − 0.471·18-s + 0.852·22-s + 2.50·23-s − 2·25-s + 1.13·28-s + 2.22·29-s − 1.06·32-s + 1/2·36-s − 3.28·37-s + 2.43·43-s − 0.904·44-s − 3.53·46-s − 3/7·49-s + 2.82·50-s − 1.06·56-s − 3.15·58-s + 0.251·63-s + 7/8·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{213444} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 213444,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9420904916$
$L(\frac12)$  $\approx$  $0.9420904916$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.958855021658912251315559727512, −8.731622022691464682737122465780, −8.013104010237496671597514230187, −7.82645545698547241323210264911, −7.31156938226239113875453585273, −6.77506565559004391547494912024, −6.46285019703242959678902869410, −5.62122186770240684074690473279, −5.14110181688607884712664879052, −4.69333471332706230755932945396, −3.77174402308413079522587358569, −3.08375157042371119503272456681, −2.37545091357255938865951638025, −1.67169823291829433403188663441, −0.792903116476162912230355051834, 0.792903116476162912230355051834, 1.67169823291829433403188663441, 2.37545091357255938865951638025, 3.08375157042371119503272456681, 3.77174402308413079522587358569, 4.69333471332706230755932945396, 5.14110181688607884712664879052, 5.62122186770240684074690473279, 6.46285019703242959678902869410, 6.77506565559004391547494912024, 7.31156938226239113875453585273, 7.82645545698547241323210264911, 8.013104010237496671597514230187, 8.731622022691464682737122465780, 8.958855021658912251315559727512

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