Properties

Degree 4
Conductor $ 2^{6} \cdot 5 \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·9-s − 2·13-s + 5·17-s − 4·25-s − 12·29-s − 8·37-s − 6·41-s − 4·45-s − 4·49-s + 12·53-s − 2·61-s + 2·65-s + 22·73-s + 7·81-s − 5·85-s − 24·89-s + 16·97-s − 12·101-s + 22·109-s − 18·113-s − 8·117-s + 14·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.447·5-s + 4/3·9-s − 0.554·13-s + 1.21·17-s − 4/5·25-s − 2.22·29-s − 1.31·37-s − 0.937·41-s − 0.596·45-s − 4/7·49-s + 1.64·53-s − 0.256·61-s + 0.248·65-s + 2.57·73-s + 7/9·81-s − 0.542·85-s − 2.54·89-s + 1.62·97-s − 1.19·101-s + 2.10·109-s − 1.69·113-s − 0.739·117-s + 1.27·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5440 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5440 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5440\)    =    \(2^{6} \cdot 5 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5440} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 5440,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.8467845197$
$L(\frac12)$  $\approx$  $0.8467845197$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;17\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−12.20534147078969727748764918038, −11.67444187629016175528015789102, −11.01683400478201041090013723110, −10.31590591645277640999898421318, −9.815333905627003028722672154773, −9.422774026925871384374309502257, −8.500583237691705298036186814207, −7.78157927400930386149956562737, −7.31406512493684766553859237503, −6.80770097490486164106618957413, −5.69224722244322325582581181803, −5.09451867572892594455543809251, −4.06065526195787235407914327840, −3.47999668677719169821330475436, −1.85357981833799781333440626914, 1.85357981833799781333440626914, 3.47999668677719169821330475436, 4.06065526195787235407914327840, 5.09451867572892594455543809251, 5.69224722244322325582581181803, 6.80770097490486164106618957413, 7.31406512493684766553859237503, 7.78157927400930386149956562737, 8.500583237691705298036186814207, 9.422774026925871384374309502257, 9.815333905627003028722672154773, 10.31590591645277640999898421318, 11.01683400478201041090013723110, 11.67444187629016175528015789102, 12.20534147078969727748764918038

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