Properties

Degree 4
Conductor $ 7^{3} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 7-s − 2·9-s − 2·13-s − 3·16-s − 6·17-s − 8·19-s + 2·25-s − 28-s + 2·36-s − 8·37-s − 6·41-s + 49-s + 2·52-s + 12·53-s − 2·61-s − 2·63-s + 7·64-s + 16·67-s + 6·68-s + 10·73-s + 8·76-s − 5·81-s − 12·83-s − 2·91-s − 2·100-s − 18·101-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.377·7-s − 2/3·9-s − 0.554·13-s − 3/4·16-s − 1.45·17-s − 1.83·19-s + 2/5·25-s − 0.188·28-s + 1/3·36-s − 1.31·37-s − 0.937·41-s + 1/7·49-s + 0.277·52-s + 1.64·53-s − 0.256·61-s − 0.251·63-s + 7/8·64-s + 1.95·67-s + 0.727·68-s + 1.17·73-s + 0.917·76-s − 5/9·81-s − 1.31·83-s − 0.209·91-s − 1/5·100-s − 1.79·101-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(41503\)    =    \(7^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{41503} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 41503,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$ \( 1 - T \)
11$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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

−10.00339241750351264117829418170, −9.361465963281960779078601642683, −8.824094859929164621480104989929, −8.473668772774342295169233493067, −8.245014544319781858814417529993, −7.15393507603149142306785349906, −6.82524420906347299159468239871, −6.31344285104076121560165509384, −5.43498540081533097817684177296, −4.93991896528081558905897577985, −4.33805413117707046904059736988, −3.78587422886686534112404405923, −2.58581886128655065343950914699, −2.03742948765031454885572790517, 0, 2.03742948765031454885572790517, 2.58581886128655065343950914699, 3.78587422886686534112404405923, 4.33805413117707046904059736988, 4.93991896528081558905897577985, 5.43498540081533097817684177296, 6.31344285104076121560165509384, 6.82524420906347299159468239871, 7.15393507603149142306785349906, 8.245014544319781858814417529993, 8.473668772774342295169233493067, 8.824094859929164621480104989929, 9.361465963281960779078601642683, 10.00339241750351264117829418170

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