Properties

Degree 4
Conductor 691
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·5-s − 7-s − 3·8-s + 2·9-s + 4·10-s + 2·11-s + 2·13-s + 14-s + 16-s − 2·18-s + 2·19-s − 4·20-s − 2·22-s + 2·25-s − 2·26-s − 28-s + 3·29-s − 2·31-s + 32-s + 4·35-s + 2·36-s − 3·37-s − 2·38-s + 12·40-s + 8·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 1.06·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s + 0.458·19-s − 0.894·20-s − 0.426·22-s + 2/5·25-s − 0.392·26-s − 0.188·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s + 0.676·35-s + 1/3·36-s − 0.493·37-s − 0.324·38-s + 1.89·40-s + 1.24·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 691 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 691 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(691\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{691} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 691,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2939463892$
$L(\frac12)$  $\approx$  $0.2939463892$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 691$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 691$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad691$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 28 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$D_{4}$ \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$V_4$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 42 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 96 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 7 T + 118 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.9020086823, −19.4142094782, −19.0743999994, −18.5125889488, −17.8340602584, −17.3776495989, −16.2849893562, −15.9762086755, −15.6646375575, −15.0524826461, −14.3209810526, −13.3902083542, −12.4759502215, −12.0343077422, −11.4304995014, −10.9092022634, −9.82605356051, −9.20536305837, −8.3903393412, −7.68684666296, −6.96753401799, −6.06606216175, −4.33714973357, −3.40174137565, 3.40174137565, 4.33714973357, 6.06606216175, 6.96753401799, 7.68684666296, 8.3903393412, 9.20536305837, 9.82605356051, 10.9092022634, 11.4304995014, 12.0343077422, 12.4759502215, 13.3902083542, 14.3209810526, 15.0524826461, 15.6646375575, 15.9762086755, 16.2849893562, 17.3776495989, 17.8340602584, 18.5125889488, 19.0743999994, 19.4142094782, 19.9020086823

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