Properties

Degree 4
Conductor $ 2^{8} \cdot 3^{4} \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  − 3·3-s − 2·5-s − 7-s + 6·9-s + 11-s + 6·13-s + 6·15-s − 10·17-s + 14·19-s + 3·21-s + 4·23-s + 5·25-s − 9·27-s + 4·29-s − 6·31-s − 3·33-s + 2·35-s + 4·37-s − 18·39-s − 3·41-s − 43-s − 12·45-s + 30·51-s + 24·53-s − 2·55-s − 42·57-s − 7·59-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s − 0.377·7-s + 2·9-s + 0.301·11-s + 1.66·13-s + 1.54·15-s − 2.42·17-s + 3.21·19-s + 0.654·21-s + 0.834·23-s + 25-s − 1.73·27-s + 0.742·29-s − 1.07·31-s − 0.522·33-s + 0.338·35-s + 0.657·37-s − 2.88·39-s − 0.468·41-s − 0.152·43-s − 1.78·45-s + 4.20·51-s + 3.29·53-s − 0.269·55-s − 5.56·57-s − 0.911·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1008} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.138363842\)
\(L(\frac12)\)  \(\approx\)  \(1.138363842\)
\(L(\frac{3}{2})\)   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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−10.36888140826630746176488953348, −9.750602134832573164493971686400, −9.390380442065762142082684866181, −9.024756521375990362504658290706, −8.437375492676719228509422488476, −8.232515500873521451287137570030, −7.23215565870312661102432643279, −7.21114181375660063017265460879, −6.77451638415183494774830382539, −6.47773240772197719564003062171, −5.72133422824476750017118463218, −5.59414197443045563636132795290, −4.88781362742784228309991891096, −4.69645324057902840504579966857, −3.80716737012656742753774995804, −3.74294154192593674966189724795, −3.02128274132811354066849985770, −2.07452330142743886164965910853, −0.906756118738555286380305438060, −0.812235298555252510886035275990, 0.812235298555252510886035275990, 0.906756118738555286380305438060, 2.07452330142743886164965910853, 3.02128274132811354066849985770, 3.74294154192593674966189724795, 3.80716737012656742753774995804, 4.69645324057902840504579966857, 4.88781362742784228309991891096, 5.59414197443045563636132795290, 5.72133422824476750017118463218, 6.47773240772197719564003062171, 6.77451638415183494774830382539, 7.21114181375660063017265460879, 7.23215565870312661102432643279, 8.232515500873521451287137570030, 8.437375492676719228509422488476, 9.024756521375990362504658290706, 9.390380442065762142082684866181, 9.750602134832573164493971686400, 10.36888140826630746176488953348

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