Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 43^{2} $
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  − 2·4-s − 4·5-s + 9-s + 6·13-s + 4·16-s − 6·17-s + 8·20-s + 2·25-s − 2·36-s + 16·37-s − 14·41-s − 4·45-s − 10·49-s − 12·52-s + 6·53-s − 16·61-s − 8·64-s − 24·65-s + 12·68-s + 24·73-s − 16·80-s + 81-s + 24·85-s + 20·89-s + 22·97-s − 4·100-s − 18·101-s + ⋯
L(s)  = 1  − 4-s − 1.78·5-s + 1/3·9-s + 1.66·13-s + 16-s − 1.45·17-s + 1.78·20-s + 2/5·25-s − 1/3·36-s + 2.63·37-s − 2.18·41-s − 0.596·45-s − 1.42·49-s − 1.66·52-s + 0.824·53-s − 2.04·61-s − 64-s − 2.97·65-s + 1.45·68-s + 2.80·73-s − 1.78·80-s + 1/9·81-s + 2.60·85-s + 2.11·89-s + 2.23·97-s − 2/5·100-s − 1.79·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 266256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(266256\)    =    \(2^{4} \cdot 3^{2} \cdot 43^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{266256} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 266256,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;43\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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

−8.510537086333408831410494114245, −8.311202822356102145576413186265, −7.80436893404074773763481968895, −7.51672561268763905855633948060, −6.78461901069866544425570526686, −6.19603689777594065734989014157, −5.96338923144874103909303632728, −4.79535808046119236916099761001, −4.74387622427760402111791858890, −4.10319001261157529164243453614, −3.50887355939377684370334646650, −3.47776319336265678852062283368, −2.16208352631894123622226457901, −1.04596298856260690087985554195, 0, 1.04596298856260690087985554195, 2.16208352631894123622226457901, 3.47776319336265678852062283368, 3.50887355939377684370334646650, 4.10319001261157529164243453614, 4.74387622427760402111791858890, 4.79535808046119236916099761001, 5.96338923144874103909303632728, 6.19603689777594065734989014157, 6.78461901069866544425570526686, 7.51672561268763905855633948060, 7.80436893404074773763481968895, 8.311202822356102145576413186265, 8.510537086333408831410494114245

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