Properties

Degree $4$
Conductor $623808$
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  − 3-s + 9-s − 4·19-s − 6·25-s − 27-s + 12·29-s − 12·41-s + 8·43-s − 14·49-s − 4·53-s + 4·57-s + 8·59-s − 4·61-s + 16·71-s + 20·73-s + 6·75-s + 81-s − 12·87-s − 12·89-s − 24·107-s + 36·113-s − 6·121-s + 12·123-s + 127-s − 8·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.917·19-s − 6/5·25-s − 0.192·27-s + 2.22·29-s − 1.87·41-s + 1.21·43-s − 2·49-s − 0.549·53-s + 0.529·57-s + 1.04·59-s − 0.512·61-s + 1.89·71-s + 2.34·73-s + 0.692·75-s + 1/9·81-s − 1.28·87-s − 1.27·89-s − 2.32·107-s + 3.38·113-s − 0.545·121-s + 1.08·123-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{623808} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.231963370\)
\(L(\frac12)\) \(\approx\) \(1.231963370\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.189315765477711565741600602856, −8.098990694093691505068092710868, −7.56166057390201911058885859268, −6.76469382211938154314381265806, −6.46690229228638557804627889575, −6.42897107072744719896577758454, −5.46401562398135459513112522798, −5.28928090027433938703648368943, −4.59390935142141694610458902429, −4.25303028692796488061253338187, −3.60315426272723356296220877606, −3.00605862781473236467551878631, −2.24389960969152461475375447009, −1.62494291573219977070674782983, −0.58501762290637151770019444985, 0.58501762290637151770019444985, 1.62494291573219977070674782983, 2.24389960969152461475375447009, 3.00605862781473236467551878631, 3.60315426272723356296220877606, 4.25303028692796488061253338187, 4.59390935142141694610458902429, 5.28928090027433938703648368943, 5.46401562398135459513112522798, 6.42897107072744719896577758454, 6.46690229228638557804627889575, 6.76469382211938154314381265806, 7.56166057390201911058885859268, 8.098990694093691505068092710868, 8.189315765477711565741600602856

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