Properties

Degree $4$
Conductor $7056$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 3·9-s − 10·25-s + 20·37-s − 16·43-s + 9·49-s − 12·63-s − 32·67-s − 8·79-s + 9·81-s − 4·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s − 40·175-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 1.51·7-s − 9-s − 2·25-s + 3.28·37-s − 2.43·43-s + 9/7·49-s − 1.51·63-s − 3.90·67-s − 0.900·79-s + 81-s − 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s − 3.02·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7056} (1, \cdot )$
Sato-Tate group: $N(\mathrm{U}(1))$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9653431326\)
\(L(\frac12)\) \(\approx\) \(0.9653431326\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.81863185793182487827709098494, −11.29182449378243979210408310932, −11.03872821307273511904902517339, −10.13000869513673846219308250556, −9.626848095956417877996067048542, −8.860431203632680285379101442976, −8.231499877023010211585730486077, −7.889190764798384171636941586833, −7.27688550516460778984917889313, −6.07005500345546038890961276394, −5.80122190357005510453887893234, −4.81237983127692909102423173270, −4.24043619370547132373815289388, −3.01248503846707493834137147349, −1.84955169590426068716106906195, 1.84955169590426068716106906195, 3.01248503846707493834137147349, 4.24043619370547132373815289388, 4.81237983127692909102423173270, 5.80122190357005510453887893234, 6.07005500345546038890961276394, 7.27688550516460778984917889313, 7.889190764798384171636941586833, 8.231499877023010211585730486077, 8.860431203632680285379101442976, 9.626848095956417877996067048542, 10.13000869513673846219308250556, 11.03872821307273511904902517339, 11.29182449378243979210408310932, 11.81863185793182487827709098494

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