Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{3} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 3·11-s + 12-s + 16-s − 9·17-s + 2·18-s + 4·19-s − 3·22-s − 24-s − 25-s − 5·27-s − 32-s + 3·33-s + 9·34-s − 2·36-s − 4·38-s − 3·41-s − 2·43-s + 3·44-s + 48-s − 4·49-s + 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 1/4·16-s − 2.18·17-s + 0.471·18-s + 0.917·19-s − 0.639·22-s − 0.204·24-s − 1/5·25-s − 0.962·27-s − 0.176·32-s + 0.522·33-s + 1.54·34-s − 1/3·36-s − 0.648·38-s − 0.468·41-s − 0.304·43-s + 0.452·44-s + 0.144·48-s − 4/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3456\)    =    \(2^{7} \cdot 3^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3456} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3456,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.6447684142$
$L(\frac12)$  $\approx$  $0.6447684142$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
3$C_2$ \( 1 - T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−12.65184638957528492562545189993, −11.76151110095666076243916527327, −11.48784336886980598015170905998, −10.96335506685343699596982946555, −10.11165591150437775767363928198, −9.458410160621127692185848356259, −8.916179542760820571893759379977, −8.556402588559235220073091676366, −7.79559470784852537439015929833, −6.94485930836828132546840783257, −6.44045207386154869417531163565, −5.48701651905825883013630977513, −4.34386851063787685085624782788, −3.28892661255911696961558158588, −2.12449507348446127913724547555, 2.12449507348446127913724547555, 3.28892661255911696961558158588, 4.34386851063787685085624782788, 5.48701651905825883013630977513, 6.44045207386154869417531163565, 6.94485930836828132546840783257, 7.79559470784852537439015929833, 8.556402588559235220073091676366, 8.916179542760820571893759379977, 9.458410160621127692185848356259, 10.11165591150437775767363928198, 10.96335506685343699596982946555, 11.48784336886980598015170905998, 11.76151110095666076243916527327, 12.65184638957528492562545189993

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