Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{4} \cdot 7^{3} $
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 − 4-s − 7-s − 3·8-s − 14-s − 16-s + 8·19-s − 6·25-s + 28-s + 4·29-s + 5·32-s + 12·37-s + 8·38-s + 49-s − 6·50-s − 12·53-s + 3·56-s + 4·58-s − 24·59-s + 7·64-s + 12·74-s − 8·76-s + 24·83-s + 98-s + 6·100-s + 16·103-s − 12·106-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s − 0.267·14-s − 1/4·16-s + 1.83·19-s − 6/5·25-s + 0.188·28-s + 0.742·29-s + 0.883·32-s + 1.97·37-s + 1.29·38-s + 1/7·49-s − 0.848·50-s − 1.64·53-s + 0.400·56-s + 0.525·58-s − 3.12·59-s + 7/8·64-s + 1.39·74-s − 0.917·76-s + 2.63·83-s + 0.101·98-s + 3/5·100-s + 1.57·103-s − 1.16·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(444528\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{444528} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 444528,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.840375511\)
\(L(\frac12)\)  \(\approx\)  \(1.840375511\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.606114381489563394279027084056, −7.906821987114438610486627525496, −7.81295430367747747098376616892, −7.30073527958403832611391139576, −6.49804351404500129234723165210, −6.06370932399531155890837002587, −5.94607665445287604157300008429, −5.02709416296405066539370067618, −4.93184851224140584283722463407, −4.22946109432700828249314767474, −3.72933010810426492886781586543, −3.05422074105458389777226041971, −2.83876601704409626449741798211, −1.70938992598143604879955116667, −0.67392178676454816034771659294, 0.67392178676454816034771659294, 1.70938992598143604879955116667, 2.83876601704409626449741798211, 3.05422074105458389777226041971, 3.72933010810426492886781586543, 4.22946109432700828249314767474, 4.93184851224140584283722463407, 5.02709416296405066539370067618, 5.94607665445287604157300008429, 6.06370932399531155890837002587, 6.49804351404500129234723165210, 7.30073527958403832611391139576, 7.81295430367747747098376616892, 7.906821987114438610486627525496, 8.606114381489563394279027084056

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