Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 31^{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  + 2·2-s + 3·4-s + 4·8-s − 6·9-s + 5·16-s − 12·18-s + 16·23-s − 6·25-s + 4·29-s + 6·32-s − 18·36-s + 20·37-s + 16·43-s + 32·46-s − 7·49-s − 12·50-s − 12·53-s + 8·58-s + 7·64-s − 24·67-s + 16·71-s − 24·72-s + 40·74-s − 16·79-s + 27·81-s + 32·86-s + 48·92-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2·9-s + 5/4·16-s − 2.82·18-s + 3.33·23-s − 6/5·25-s + 0.742·29-s + 1.06·32-s − 3·36-s + 3.28·37-s + 2.43·43-s + 4.71·46-s − 49-s − 1.69·50-s − 1.64·53-s + 1.05·58-s + 7/8·64-s − 2.93·67-s + 1.89·71-s − 2.82·72-s + 4.64·74-s − 1.80·79-s + 3·81-s + 3.45·86-s + 5.00·92-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 188356 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 188356 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(188356\)    =    \(2^{2} \cdot 7^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{188356} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 188356,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.753323965$
$L(\frac12)$  $\approx$  $3.753323965$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;31\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + p T^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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

−9.056498267458590117851162621111, −8.767717911672973124583558991985, −7.85520408569034144174211600673, −7.76857292337562945056886541715, −7.13164735275382460400374069877, −6.32858579420662158347624019611, −6.10481334469888763444585228494, −5.73966246913035569800644487489, −4.99493437014528324856589187903, −4.74203912193695697715499891246, −4.05486645798658988472696574741, −3.16188554299090604324575557253, −2.88035578683009317838785569193, −2.44397171990298187609757473706, −1.04640991976554111715719591186, 1.04640991976554111715719591186, 2.44397171990298187609757473706, 2.88035578683009317838785569193, 3.16188554299090604324575557253, 4.05486645798658988472696574741, 4.74203912193695697715499891246, 4.99493437014528324856589187903, 5.73966246913035569800644487489, 6.10481334469888763444585228494, 6.32858579420662158347624019611, 7.13164735275382460400374069877, 7.76857292337562945056886541715, 7.85520408569034144174211600673, 8.767717911672973124583558991985, 9.056498267458590117851162621111

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