Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{4} \cdot 167^{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 + 3·7-s + 9·13-s − 17-s + 4·19-s + 23-s + 17·25-s − 8·29-s + 6·31-s + 18·35-s + 8·37-s − 2·41-s + 12·43-s + 4·47-s − 4·49-s + 2·53-s + 6·59-s − 8·61-s + 54·65-s + 2·67-s + 3·71-s − 17·73-s + 20·79-s − 4·83-s − 6·85-s + 6·89-s + 27·91-s + ⋯
L(s)  = 1  + 2.68·5-s + 1.13·7-s + 2.49·13-s − 0.242·17-s + 0.917·19-s + 0.208·23-s + 17/5·25-s − 1.48·29-s + 1.07·31-s + 3.04·35-s + 1.31·37-s − 0.312·41-s + 1.82·43-s + 0.583·47-s − 4/7·49-s + 0.274·53-s + 0.781·59-s − 1.02·61-s + 6.69·65-s + 0.244·67-s + 0.356·71-s − 1.98·73-s + 2.25·79-s − 0.439·83-s − 0.650·85-s + 0.635·89-s + 2.83·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36144144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36144144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36144144\)    =    \(2^{4} \cdot 3^{4} \cdot 167^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6012} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 36144144,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $10.26466291$
$L(\frac12)$  $\approx$  $10.26466291$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3 \( 1 \)
167$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 31 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 77 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 85 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 21 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 115 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 17 T + 189 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 20 T + 245 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 157 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 133 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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

−8.187282186472679021982844735090, −8.101739136351671933754826703925, −7.44346826825739842210961240083, −7.31744006795061071169621830091, −6.47718426162445350139061455084, −6.47512391492057174641361757289, −5.98879049735523798975252584910, −5.85936947209048348045760151659, −5.39367056971665751879451724818, −5.36762365329716937398940319490, −4.53044945036982782249640413760, −4.51976553544408990747927626230, −3.67165279116746009034227060832, −3.58959185030604405792808054236, −2.64463515503432810562383424323, −2.62723449833689006743733027752, −1.89342785702541234455822388234, −1.66328674420939585085062941828, −1.17500671843368084589159178289, −0.910994184126424918935185357619, 0.910994184126424918935185357619, 1.17500671843368084589159178289, 1.66328674420939585085062941828, 1.89342785702541234455822388234, 2.62723449833689006743733027752, 2.64463515503432810562383424323, 3.58959185030604405792808054236, 3.67165279116746009034227060832, 4.51976553544408990747927626230, 4.53044945036982782249640413760, 5.36762365329716937398940319490, 5.39367056971665751879451724818, 5.85936947209048348045760151659, 5.98879049735523798975252584910, 6.47512391492057174641361757289, 6.47718426162445350139061455084, 7.31744006795061071169621830091, 7.44346826825739842210961240083, 8.101739136351671933754826703925, 8.187282186472679021982844735090

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