Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 19^{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 + 3·7-s + 4·8-s − 5·9-s + 4·11-s + 6·14-s + 5·16-s − 10·18-s + 8·22-s − 2·23-s + 6·25-s + 9·28-s − 10·29-s + 6·32-s − 15·36-s − 4·37-s + 8·43-s + 12·44-s − 4·46-s + 2·49-s + 12·50-s − 2·53-s + 12·56-s − 20·58-s − 15·63-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.13·7-s + 1.41·8-s − 5/3·9-s + 1.20·11-s + 1.60·14-s + 5/4·16-s − 2.35·18-s + 1.70·22-s − 0.417·23-s + 6/5·25-s + 1.70·28-s − 1.85·29-s + 1.06·32-s − 5/2·36-s − 0.657·37-s + 1.21·43-s + 1.80·44-s − 0.589·46-s + 2/7·49-s + 1.69·50-s − 0.274·53-s + 1.60·56-s − 2.62·58-s − 1.88·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 70756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 70756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(70756\)    =    \(2^{2} \cdot 7^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{70756} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 70756,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.647568683$
$L(\frac12)$  $\approx$  $3.647568683$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 - 3 T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−9.856310223818109751782129963913, −9.254038079040445697892143439151, −8.655339283587140731226502868667, −8.408486909011940238981299566497, −7.60323894240594246780554900498, −7.25038064772684706053554512263, −6.45399284641636243911038304495, −6.09315132100286987427486982952, −5.29912825884459151712444657250, −5.28069074518443602737935578596, −4.32145140400317761773094051986, −3.86057015395193111701613139713, −3.13458058385460483521784962114, −2.39336202780458240475893030948, −1.50913345329536755396223442365, 1.50913345329536755396223442365, 2.39336202780458240475893030948, 3.13458058385460483521784962114, 3.86057015395193111701613139713, 4.32145140400317761773094051986, 5.28069074518443602737935578596, 5.29912825884459151712444657250, 6.09315132100286987427486982952, 6.45399284641636243911038304495, 7.25038064772684706053554512263, 7.60323894240594246780554900498, 8.408486909011940238981299566497, 8.655339283587140731226502868667, 9.254038079040445697892143439151, 9.856310223818109751782129963913

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