Properties

Degree 4
Conductor $ 5 \cdot 277 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 7-s − 4·8-s + 2·9-s + 4·10-s − 6·12-s − 13-s + 2·14-s + 6·15-s + 8·16-s − 4·18-s + 19-s − 4·20-s + 3·21-s + 23-s + 12·24-s − 2·25-s + 2·26-s + 6·27-s − 2·28-s − 5·29-s − 12·30-s − 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.73·12-s − 0.277·13-s + 0.534·14-s + 1.54·15-s + 2·16-s − 0.942·18-s + 0.229·19-s − 0.894·20-s + 0.654·21-s + 0.208·23-s + 2.44·24-s − 2/5·25-s + 0.392·26-s + 1.15·27-s − 0.377·28-s − 0.928·29-s − 2.19·30-s − 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1385 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1385 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1385\)    =    \(5 \cdot 277\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1385} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1385,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;277\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;277\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
277$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 24 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 3 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 11 T + 152 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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

−19.6273643581, −18.8639617108, −18.5671888805, −18.0220939371, −17.4836945764, −16.9995768034, −16.8168020807, −16.0736788406, −15.5842115729, −15.0360764322, −14.2963698968, −13.1987788447, −12.3245514773, −12.0211525907, −11.4317925881, −11.0731814029, −10.2666941419, −9.63078676148, −8.91530033341, −8.16968598156, −7.42758039961, −6.47110463922, −5.90223324063, −5.00367531507, −3.36744843913, 0, 3.36744843913, 5.00367531507, 5.90223324063, 6.47110463922, 7.42758039961, 8.16968598156, 8.91530033341, 9.63078676148, 10.2666941419, 11.0731814029, 11.4317925881, 12.0211525907, 12.3245514773, 13.1987788447, 14.2963698968, 15.0360764322, 15.5842115729, 16.0736788406, 16.8168020807, 16.9995768034, 17.4836945764, 18.0220939371, 18.5671888805, 18.8639617108, 19.6273643581

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