Properties

 Degree 4 Conductor $5^{2} \cdot 41$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

Origins

Dirichlet series

 L(s)  = 1 − 2·4-s − 5-s − 4·9-s + 3·11-s + 5·19-s + 2·20-s − 4·25-s + 11·29-s − 6·31-s + 8·36-s − 41-s − 6·44-s + 4·45-s − 6·49-s − 3·55-s + 59-s − 2·61-s + 8·64-s − 3·71-s − 10·76-s − 15·79-s + 7·81-s − 9·89-s − 5·95-s − 12·99-s + 8·100-s + 15·101-s + ⋯
 L(s)  = 1 − 4-s − 0.447·5-s − 4/3·9-s + 0.904·11-s + 1.14·19-s + 0.447·20-s − 4/5·25-s + 2.04·29-s − 1.07·31-s + 4/3·36-s − 0.156·41-s − 0.904·44-s + 0.596·45-s − 6/7·49-s − 0.404·55-s + 0.130·59-s − 0.256·61-s + 64-s − 0.356·71-s − 1.14·76-s − 1.68·79-s + 7/9·81-s − 0.953·89-s − 0.512·95-s − 1.20·99-s + 4/5·100-s + 1.49·101-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$4$$ $$N$$ = $$1025$$    =    $$5^{2} \cdot 41$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{1025} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 1025,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.4222141590$ $L(\frac12)$ $\approx$ $0.4222141590$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;41\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;41\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ $$1 + T + p T^{2}$$
41$C_1$$\times$$C_2$ $$( 1 + T )( 1 + p T^{2} )$$
good2$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
3$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
7$C_2^2$ $$1 + 6 T^{2} + p^{2} T^{4}$$
11$C_2$$\times$$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
13$C_2^2$ $$1 - 16 T^{2} + p^{2} T^{4}$$
17$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} )$$
23$C_2^2$ $$1 - 19 T^{2} + p^{2} T^{4}$$
29$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
31$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
37$C_2^2$ $$1 - 61 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - p T^{2} )^{2}$$
47$C_2^2$ $$1 - 40 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 74 T^{2} + p^{2} T^{4}$$
59$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
61$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
67$C_2^2$ $$1 + 88 T^{2} + p^{2} T^{4}$$
71$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
73$C_2^2$ $$1 + 109 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
83$C_2^2$ $$1 - 79 T^{2} + p^{2} T^{4}$$
89$C_2$$\times$$C_2$ $$( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} )$$
97$C_2^2$ $$1 + 152 T^{2} + p^{2} T^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}