Properties

Degree 4
Conductor $ 2^{7} \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  − 2-s + 4-s + 2·7-s − 8-s − 2·9-s − 2·14-s + 16-s + 12·17-s + 2·18-s − 10·25-s + 2·28-s − 8·31-s − 32-s − 12·34-s − 2·36-s + 12·41-s − 24·47-s + 3·49-s + 10·50-s − 2·56-s + 8·62-s − 4·63-s + 64-s + 12·68-s + 2·72-s + 4·73-s + 16·79-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.534·14-s + 1/4·16-s + 2.91·17-s + 0.471·18-s − 2·25-s + 0.377·28-s − 1.43·31-s − 0.176·32-s − 2.05·34-s − 1/3·36-s + 1.87·41-s − 3.50·47-s + 3/7·49-s + 1.41·50-s − 0.267·56-s + 1.01·62-s − 0.503·63-s + 1/8·64-s + 1.45·68-s + 0.235·72-s + 0.468·73-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6272\)    =    \(2^{7} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6272} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 6272,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.6939750917$
$L(\frac12)$  $\approx$  $0.6939750917$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−11.96768802253889907265831052426, −11.23136141438460806904855801814, −11.06714116909225412617528172908, −10.04491938039307927961472194001, −9.765547119459919407856461234632, −9.195409247501138715663586603039, −8.222985047096640302551257734717, −7.87021772021814993834479658639, −7.57571100088867902110310233811, −6.47781377385923123584313505983, −5.57928681742950427486583645839, −5.36663802373794330843690451326, −3.93522119575008455702519349301, −3.07302685938302839530976846531, −1.64950502134773404742708641242, 1.64950502134773404742708641242, 3.07302685938302839530976846531, 3.93522119575008455702519349301, 5.36663802373794330843690451326, 5.57928681742950427486583645839, 6.47781377385923123584313505983, 7.57571100088867902110310233811, 7.87021772021814993834479658639, 8.222985047096640302551257734717, 9.195409247501138715663586603039, 9.765547119459919407856461234632, 10.04491938039307927961472194001, 11.06714116909225412617528172908, 11.23136141438460806904855801814, 11.96768802253889907265831052426

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