Properties

 Degree 4 Conductor $3^{4} \cdot 5^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

Origins

Dirichlet series

 L(s)  = 1 − 4-s − 3·16-s − 8·19-s − 5·25-s + 16·31-s + 14·49-s + 4·61-s + 7·64-s + 8·76-s − 32·79-s + 5·100-s + 28·109-s − 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯
 L(s)  = 1 − 1/2·4-s − 3/4·16-s − 1.83·19-s − 25-s + 2.87·31-s + 2·49-s + 0.512·61-s + 7/8·64-s + 0.917·76-s − 3.60·79-s + 1/2·100-s + 2.68·109-s − 2·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$4$$ $$N$$ = $$2025$$    =    $$3^{4} \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{2025} (1, \cdot )$ Sato-Tate : $N(\mathrm{U}(1))$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(4,\ 2025,\ (\ :1/2, 1/2),\ 1)$ $L(1)$ $\approx$ $0.5869732169$ $L(\frac12)$ $\approx$ $0.5869732169$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;5\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3 $$1$$
5$C_2$ $$1 + p T^{2}$$
good2$V_4$ $$1 + T^{2} + p^{2} T^{4}$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
11$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - p T^{2} )^{2}$$
17$V_4$ $$1 - 14 T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
23$V_4$ $$1 + 34 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
37$C_2$ $$( 1 - p T^{2} )^{2}$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 - p T^{2} )^{2}$$
47$V_4$ $$1 - 14 T^{2} + p^{2} T^{4}$$
53$V_4$ $$1 - 86 T^{2} + p^{2} T^{4}$$
59$C_2$ $$( 1 + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - p T^{2} )^{2}$$
71$C_2$ $$( 1 + p T^{2} )^{2}$$
73$C_2$ $$( 1 - p T^{2} )^{2}$$
79$C_2$ $$( 1 + 16 T + p T^{2} )^{2}$$
83$V_4$ $$1 + 154 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 + p T^{2} )^{2}$$
97$C_2$ $$( 1 - p T^{2} )^{2}$$
\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}