Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·5-s + 3·9-s − 11-s − 8·15-s − 16·23-s + 2·25-s − 4·27-s + 2·33-s − 12·37-s + 12·45-s − 10·49-s + 28·53-s − 4·55-s − 24·59-s + 8·67-s + 32·69-s − 4·75-s + 5·81-s − 28·89-s − 4·97-s − 3·99-s − 24·103-s + 24·111-s − 20·113-s − 64·115-s + 121-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 9-s − 0.301·11-s − 2.06·15-s − 3.33·23-s + 2/5·25-s − 0.769·27-s + 0.348·33-s − 1.97·37-s + 1.78·45-s − 1.42·49-s + 3.84·53-s − 0.539·55-s − 3.12·59-s + 0.977·67-s + 3.85·69-s − 0.461·75-s + 5/9·81-s − 2.96·89-s − 0.406·97-s − 0.301·99-s − 2.36·103-s + 2.27·111-s − 1.88·113-s − 5.96·115-s + 1/11·121-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(191664\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{191664} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 191664,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.106296123632974974434793663066, −8.281330370411433339491186890517, −8.017674089261979349140094913788, −7.32585666682316342415587434616, −6.63133552101688268434660185897, −6.41893627610765861044252902088, −5.68129524329839044399202288761, −5.62860857478680063008223727075, −5.24185435258333597768937468982, −4.18020563951979381603420573914, −4.01176346711548250450125288086, −2.82857722936746475093153693417, −1.86785039249087723872477729694, −1.74190329399839288958892267266, 0, 1.74190329399839288958892267266, 1.86785039249087723872477729694, 2.82857722936746475093153693417, 4.01176346711548250450125288086, 4.18020563951979381603420573914, 5.24185435258333597768937468982, 5.62860857478680063008223727075, 5.68129524329839044399202288761, 6.41893627610765861044252902088, 6.63133552101688268434660185897, 7.32585666682316342415587434616, 8.017674089261979349140094913788, 8.281330370411433339491186890517, 9.106296123632974974434793663066

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