Properties

Degree $4$
Conductor $9216$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 4·17-s + 16·23-s − 6·25-s − 16·31-s − 12·41-s − 14·49-s − 16·71-s + 20·73-s + 16·79-s + 81-s − 12·89-s + 4·97-s − 32·103-s + 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s − 22·169-s + ⋯
L(s)  = 1  + 1/3·9-s + 0.970·17-s + 3.33·23-s − 6/5·25-s − 2.87·31-s − 1.87·41-s − 2·49-s − 1.89·71-s + 2.34·73-s + 1.80·79-s + 1/9·81-s − 1.27·89-s + 0.406·97-s − 3.15·103-s + 3.38·113-s − 0.545·121-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 + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9216} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9216,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.004711853\)
\(L(\frac12)\) \(\approx\) \(1.004711853\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.45788576598466513578332775969, −11.02059068705223596291987853205, −10.68009321996461187722201460512, −9.656797755147918818405236990829, −9.549120050570062839357860330869, −8.777512613188475551383845278968, −8.173282349834764667604075089133, −7.26958585488936859387169650645, −7.13146007277855219299925953915, −6.21040581024747445781537308376, −5.24179959934618595033608799542, −5.01357758335939619070346679818, −3.71523857372513194753595622108, −3.14826068230388029805668446658, −1.64229196717434907696557687175, 1.64229196717434907696557687175, 3.14826068230388029805668446658, 3.71523857372513194753595622108, 5.01357758335939619070346679818, 5.24179959934618595033608799542, 6.21040581024747445781537308376, 7.13146007277855219299925953915, 7.26958585488936859387169650645, 8.173282349834764667604075089133, 8.777512613188475551383845278968, 9.549120050570062839357860330869, 9.656797755147918818405236990829, 10.68009321996461187722201460512, 11.02059068705223596291987853205, 11.45788576598466513578332775969

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