Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{6} \cdot 7^{2} $
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  + 6·5-s − 4·7-s − 12·17-s + 17·25-s − 24·35-s + 2·37-s − 6·41-s − 20·43-s + 12·47-s + 9·49-s − 12·59-s − 4·67-s − 28·79-s − 12·83-s − 72·85-s + 18·89-s + 36·101-s − 22·109-s + 48·119-s − 5·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2.68·5-s − 1.51·7-s − 2.91·17-s + 17/5·25-s − 4.05·35-s + 0.328·37-s − 0.937·41-s − 3.04·43-s + 1.75·47-s + 9/7·49-s − 1.56·59-s − 0.488·67-s − 3.15·79-s − 1.31·83-s − 7.80·85-s + 1.90·89-s + 3.58·101-s − 2.10·109-s + 4.40·119-s − 0.454·121-s + 1.60·125-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3024} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.542462125\)
\(L(\frac12)\)  \(\approx\)  \(1.542462125\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 115 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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

−8.925613923807683273725080869698, −8.657458522502953569830698386143, −8.610190444332558145458579339150, −7.66888029502150422230762208426, −7.21430312976100682884341875726, −6.71640111409935872663977329888, −6.61629398488388570737120462336, −6.28127445935905180683440789933, −5.98826615237472315433458688777, −5.61578839880169822821976620403, −5.20919025650180824820006485366, −4.53735762689570983327516331494, −4.50371149494191360826947695984, −3.65667840947889635022657714151, −3.16833700260265901994212883162, −2.64552566914310812396839325346, −2.37577536382199931415664425892, −1.74076340317028418083910345390, −1.61341195233487171607866973412, −0.34832893972256891121407140849, 0.34832893972256891121407140849, 1.61341195233487171607866973412, 1.74076340317028418083910345390, 2.37577536382199931415664425892, 2.64552566914310812396839325346, 3.16833700260265901994212883162, 3.65667840947889635022657714151, 4.50371149494191360826947695984, 4.53735762689570983327516331494, 5.20919025650180824820006485366, 5.61578839880169822821976620403, 5.98826615237472315433458688777, 6.28127445935905180683440789933, 6.61629398488388570737120462336, 6.71640111409935872663977329888, 7.21430312976100682884341875726, 7.66888029502150422230762208426, 8.610190444332558145458579339150, 8.657458522502953569830698386143, 8.925613923807683273725080869698

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