Properties

Degree 4
Conductor $ 2^{5} \cdot 691 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 4·5-s − 4·7-s + 7·9-s − 4·11-s − 13-s + 16·15-s − 17-s − 8·19-s + 16·21-s − 23-s + 3·25-s − 4·27-s − 29-s + 31-s + 16·33-s + 16·35-s + 7·37-s + 4·39-s + 5·41-s − 6·43-s − 28·45-s + 4·47-s + 3·49-s + 4·51-s − 15·53-s + 16·55-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.78·5-s − 1.51·7-s + 7/3·9-s − 1.20·11-s − 0.277·13-s + 4.13·15-s − 0.242·17-s − 1.83·19-s + 3.49·21-s − 0.208·23-s + 3/5·25-s − 0.769·27-s − 0.185·29-s + 0.179·31-s + 2.78·33-s + 2.70·35-s + 1.15·37-s + 0.640·39-s + 0.780·41-s − 0.914·43-s − 4.17·45-s + 0.583·47-s + 3/7·49-s + 0.560·51-s − 2.06·53-s + 2.15·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(22112\)    =    \(2^{5} \cdot 691\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{22112} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 22112,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;691\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;691\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
691$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 38 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T - 49 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 40 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 15 T + 121 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T - 21 T^{2} + p T^{3} + p^{2} T^{4} \)
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

−16.0500424610, −15.8351235472, −15.5101957298, −14.9897715596, −14.3698371963, −13.2484281159, −13.1290529901, −12.4975392187, −12.2099396995, −11.8701716449, −11.1957820433, −10.8377837457, −10.7192700793, −9.85989513158, −9.44793568370, −8.38393961412, −7.97846575436, −7.35390365950, −6.65324845722, −6.20893369004, −5.84171526201, −4.94013524184, −4.43903540517, −3.73618041581, −2.73130520401, 0, 0, 2.73130520401, 3.73618041581, 4.43903540517, 4.94013524184, 5.84171526201, 6.20893369004, 6.65324845722, 7.35390365950, 7.97846575436, 8.38393961412, 9.44793568370, 9.85989513158, 10.7192700793, 10.8377837457, 11.1957820433, 11.8701716449, 12.2099396995, 12.4975392187, 13.1290529901, 13.2484281159, 14.3698371963, 14.9897715596, 15.5101957298, 15.8351235472, 16.0500424610

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