Properties

Degree 4
Conductor $ 5^{2} \cdot 197 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s − 3·5-s + 8·6-s − 5·7-s + 4·8-s + 7·9-s + 6·10-s − 2·11-s − 13-s + 10·14-s + 12·15-s − 4·16-s − 2·17-s − 14·18-s − 4·19-s + 20·21-s + 4·22-s − 16·24-s + 4·25-s + 2·26-s − 4·27-s − 5·29-s − 24·30-s − 2·31-s + 8·33-s + 4·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s − 1.34·5-s + 3.26·6-s − 1.88·7-s + 1.41·8-s + 7/3·9-s + 1.89·10-s − 0.603·11-s − 0.277·13-s + 2.67·14-s + 3.09·15-s − 16-s − 0.485·17-s − 3.29·18-s − 0.917·19-s + 4.36·21-s + 0.852·22-s − 3.26·24-s + 4/5·25-s + 0.392·26-s − 0.769·27-s − 0.928·29-s − 4.38·30-s − 0.359·31-s + 1.39·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4925\)    =    \(5^{2} \cdot 197\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4925} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 4925,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;197\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;197\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2$ \( 1 + 3 T + p T^{2} \)
197$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 18 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 2 T - 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 37 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 11 T + 115 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 103 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 11 T + 171 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
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

−18.0950397109, −17.2769531254, −17.2110240745, −16.5585618838, −16.4138296994, −15.8587759136, −15.4195683337, −14.7254396084, −13.5720923603, −12.9818977495, −12.7149246391, −12.0350508689, −11.5634488848, −11.0015156675, −10.4402295637, −10.0957416573, −9.39847415589, −8.77630676872, −8.08411604947, −7.30687105402, −6.65177386822, −6.11272827742, −5.20160658195, −4.49571366313, −3.46735677122, 0, 0, 3.46735677122, 4.49571366313, 5.20160658195, 6.11272827742, 6.65177386822, 7.30687105402, 8.08411604947, 8.77630676872, 9.39847415589, 10.0957416573, 10.4402295637, 11.0015156675, 11.5634488848, 12.0350508689, 12.7149246391, 12.9818977495, 13.5720923603, 14.7254396084, 15.4195683337, 15.8587759136, 16.4138296994, 16.5585618838, 17.2110240745, 17.2769531254, 18.0950397109

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