Properties

Degree 4
Conductor $ 3^{2} \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s − 2·7-s + 9-s + 2·12-s + 4·13-s − 3·16-s + 4·19-s + 4·21-s + 2·25-s + 4·27-s + 2·28-s − 14·31-s − 36-s + 4·37-s − 8·39-s − 2·43-s + 6·48-s − 2·49-s − 4·52-s − 8·57-s − 8·61-s − 2·63-s + 7·64-s + 4·67-s + 3·73-s − 4·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.577·12-s + 1.10·13-s − 3/4·16-s + 0.917·19-s + 0.872·21-s + 2/5·25-s + 0.769·27-s + 0.377·28-s − 2.51·31-s − 1/6·36-s + 0.657·37-s − 1.28·39-s − 0.304·43-s + 0.866·48-s − 2/7·49-s − 0.554·52-s − 1.05·57-s − 1.02·61-s − 0.251·63-s + 7/8·64-s + 0.488·67-s + 0.351·73-s − 0.461·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 657 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 657 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(657\)    =    \(3^{2} \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{657} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 657,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3119937432$
$L(\frac12)$  $\approx$  $0.3119937432$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;73\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;73\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
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

−14.84173356019134395733330207486, −14.08580238690987163680420223855, −13.41393326234914558128432992383, −12.86008664859908357365209131594, −12.24709854705253803421459273792, −11.23211640494482195728770394702, −11.11982146179022989852749019370, −10.11776565657729148299697331717, −9.279388737201940931926713800549, −8.726859407077604520825403564537, −7.47402283910884401831234750195, −6.54100615181400697590529104632, −5.81111174203705729174975290829, −4.91652301905022788110545307688, −3.56521670106926636818196449033, 3.56521670106926636818196449033, 4.91652301905022788110545307688, 5.81111174203705729174975290829, 6.54100615181400697590529104632, 7.47402283910884401831234750195, 8.726859407077604520825403564537, 9.279388737201940931926713800549, 10.11776565657729148299697331717, 11.11982146179022989852749019370, 11.23211640494482195728770394702, 12.24709854705253803421459273792, 12.86008664859908357365209131594, 13.41393326234914558128432992383, 14.08580238690987163680420223855, 14.84173356019134395733330207486

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