Properties

Degree 4
Conductor $ 7^{2} \cdot 11^{3} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 5·9-s + 11-s + 8·13-s − 4·16-s − 4·17-s − 2·23-s − 9·25-s + 6·37-s − 16·41-s − 3·49-s − 12·53-s + 24·61-s + 10·63-s − 14·67-s − 6·71-s + 8·73-s − 2·77-s + 16·81-s − 12·83-s − 16·91-s − 5·99-s + 4·101-s + 8·112-s + 18·113-s − 40·117-s + 8·119-s + ⋯
L(s)  = 1  − 0.755·7-s − 5/3·9-s + 0.301·11-s + 2.21·13-s − 16-s − 0.970·17-s − 0.417·23-s − 9/5·25-s + 0.986·37-s − 2.49·41-s − 3/7·49-s − 1.64·53-s + 3.07·61-s + 1.25·63-s − 1.71·67-s − 0.712·71-s + 0.936·73-s − 0.227·77-s + 16/9·81-s − 1.31·83-s − 1.67·91-s − 0.502·99-s + 0.398·101-s + 0.755·112-s + 1.69·113-s − 3.69·117-s + 0.733·119-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(65219\)    =    \(7^{2} \cdot 11^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{65219} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 65219,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−9.635355093326247349773689827545, −8.982234715263527510913855271701, −8.603539619290756001226038948684, −8.383354580077423849355753624090, −7.73211325750284141433186231976, −6.81130354135159597032281026635, −6.36261389471308870138602900888, −6.10008461377510957946279965770, −5.56513823603813445068893148660, −4.72966200734041708156501051889, −3.84763496937519983566731282131, −3.52931135642754093199166167311, −2.68793343495993615580735780031, −1.77464628596756662135858026846, 0, 1.77464628596756662135858026846, 2.68793343495993615580735780031, 3.52931135642754093199166167311, 3.84763496937519983566731282131, 4.72966200734041708156501051889, 5.56513823603813445068893148660, 6.10008461377510957946279965770, 6.36261389471308870138602900888, 6.81130354135159597032281026635, 7.73211325750284141433186231976, 8.383354580077423849355753624090, 8.603539619290756001226038948684, 8.982234715263527510913855271701, 9.635355093326247349773689827545

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