Properties

Degree 4
Conductor $ 7^{2} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 2·7-s − 8·8-s − 5·9-s + 2·11-s + 8·14-s − 4·16-s + 20·18-s − 8·22-s − 2·23-s − 9·25-s − 16·28-s + 32·32-s − 40·36-s + 6·37-s − 12·43-s + 16·44-s + 8·46-s − 3·49-s + 36·50-s − 12·53-s + 16·56-s + 10·63-s − 64·64-s − 14·67-s − 6·71-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 0.755·7-s − 2.82·8-s − 5/3·9-s + 0.603·11-s + 2.13·14-s − 16-s + 4.71·18-s − 1.70·22-s − 0.417·23-s − 9/5·25-s − 3.02·28-s + 5.65·32-s − 6.66·36-s + 0.986·37-s − 1.82·43-s + 2.41·44-s + 1.17·46-s − 3/7·49-s + 5.09·50-s − 1.64·53-s + 2.13·56-s + 1.25·63-s − 8·64-s − 1.71·67-s − 0.712·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5929\)    =    \(7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5929} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 5929,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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

−11.45125861034521058353374083886, −11.12654606872452134010733513837, −10.03550909718107888433464868208, −10.03359978338253814456351958606, −9.410787129321073286385580378997, −8.678914216536525529374637909269, −8.603539619290756001226038948684, −7.77729309499855305690824525396, −7.38142851192821190173908719125, −6.36261389471308870138602900888, −5.98958560322778025003846316474, −4.56768346065918811492983607197, −3.18179592020545258264561636452, −1.88740391319498359233714309471, 0, 1.88740391319498359233714309471, 3.18179592020545258264561636452, 4.56768346065918811492983607197, 5.98958560322778025003846316474, 6.36261389471308870138602900888, 7.38142851192821190173908719125, 7.77729309499855305690824525396, 8.603539619290756001226038948684, 8.678914216536525529374637909269, 9.410787129321073286385580378997, 10.03359978338253814456351958606, 10.03550909718107888433464868208, 11.12654606872452134010733513837, 11.45125861034521058353374083886

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