Properties

Degree 4
Conductor $ 2 \cdot 7^{2} \cdot 79 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 5·4-s − 7-s − 5·8-s − 9-s − 9·11-s + 3·14-s + 16-s + 3·18-s + 27·22-s − 7·23-s − 3·25-s − 5·28-s − 29-s + 7·32-s − 5·36-s + 14·37-s − 6·43-s − 45·44-s + 21·46-s − 6·49-s + 9·50-s − 53-s + 5·56-s + 3·58-s + 63-s − 15·64-s + ⋯
L(s)  = 1  − 2.12·2-s + 5/2·4-s − 0.377·7-s − 1.76·8-s − 1/3·9-s − 2.71·11-s + 0.801·14-s + 1/4·16-s + 0.707·18-s + 5.75·22-s − 1.45·23-s − 3/5·25-s − 0.944·28-s − 0.185·29-s + 1.23·32-s − 5/6·36-s + 2.30·37-s − 0.914·43-s − 6.78·44-s + 3.09·46-s − 6/7·49-s + 1.27·50-s − 0.137·53-s + 0.668·56-s + 0.393·58-s + 0.125·63-s − 1.87·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7742\)    =    \(2 \cdot 7^{2} \cdot 79\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{7742} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 7742,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;79\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;79\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T + p T^{2} ) \)
7$C_2$ \( 1 + T + p T^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 11 T + p T^{2} ) \)
good3$V_4$ \( 1 + T^{2} + p^{2} T^{4} \)
5$V_4$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$V_4$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$V_4$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
19$V_4$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$V_4$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
41$V_4$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$V_4$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
59$V_4$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
61$V_4$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
73$V_4$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
89$V_4$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
97$V_4$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
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

−11.21366454432362178409800896144, −10.67031227449828284600588290559, −10.15719752606998884328845703933, −9.827750040154990690504559207296, −9.348106052753687446219355176798, −8.391620093074762865941497641656, −8.056074514521538854040464618150, −7.77684157811220129973739110796, −7.07843402085020079516919923744, −6.12517083143332380413032407232, −5.51689714943361794564222825282, −4.48039111841836744764506978027, −2.94840603147446336550026274361, −2.15224515309300368213938566990, 0, 2.15224515309300368213938566990, 2.94840603147446336550026274361, 4.48039111841836744764506978027, 5.51689714943361794564222825282, 6.12517083143332380413032407232, 7.07843402085020079516919923744, 7.77684157811220129973739110796, 8.056074514521538854040464618150, 8.391620093074762865941497641656, 9.348106052753687446219355176798, 9.827750040154990690504559207296, 10.15719752606998884328845703933, 10.67031227449828284600588290559, 11.21366454432362178409800896144

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