# Properties

 Degree 4 Conductor 277 Sign $1$ Self-dual yes Motivic weight 1

# Related objects

## Dirichlet series

 L(A,s)  = 1 − 2·2-s − 3-s − 5-s + 2·6-s + 7-s + 4·8-s + 2·10-s − 2·11-s + 3·13-s − 2·14-s + 15-s − 4·16-s − 4·17-s − 19-s − 21-s + 4·22-s + 3·23-s − 4·24-s + 3·25-s − 6·26-s − 2·27-s − 29-s − 2·30-s − 10·31-s + 2·33-s + 8·34-s − 35-s + ⋯
 L(s,A)  = 1 − 1.414·2-s − 0.577·3-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1.414·8-s + 0.632·10-s − 0.603·11-s + 0.832·13-s − 0.534·14-s + 0.258·15-s − 16-s − 0.970·17-s − 0.229·19-s − 0.218·21-s + 0.852·22-s + 0.625·23-s − 0.816·24-s + 0.6·25-s − 1.176·26-s − 0.384·27-s − 0.185·29-s − 0.365·30-s − 1.796·31-s + 0.348·33-s + 1.371·34-s − 0.169·35-s + ⋯

## Functional equation

\begin{align} \Lambda(A,s)=\mathstrut & 277 ^{s/2} \Gamma_{\C}(s) ^{2} \cdot L(A,s)\cr =\mathstrut & \Lambda(A, 2-s) \end{align}
\begin{align} \Lambda(s,A)=\mathstrut & 277 ^{s/2} \Gamma_{\C}(s+0.5) ^{2} \cdot L(s,A)\cr =\mathstrut & \Lambda(1-s,A) \end{align}

## Invariants

 $d$ = $4$ $N$ = $277$ $\varepsilon$ = $1$ weight = 1 character : $\chi_{277} (1, \cdot )$ Sato-Tate : $\mathrm{USp}(4)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(4,\ 277,\ (\ :1/2, 1/2),\ 1)$ $L(A,1)$ $\approx$ $0.1431366606$ $L(\frac12,A)$ $\approx$ $0.1431366606$ $L(A,\frac{3}{2})$ not available $L(1,A)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 277$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p = 277$, then $F_p$ is a polynomial of degree at most 3.
$p$$F_p$$\Gal(F_p)$
bad277$(1+T)(1-8T+277T^{2})$$C_1$$\times$$C_2 good2(1+2T^{2})(1+2T+2T^{2})$$C_2$$\times$$C_2$
3$1+T+T^{2}+3T^{3}+9T^{4}$$D_4 5(1-3T+5T^{2})(1+4T+5T^{2})$$C_2$$\times$$C_2$
7$1-T+3T^{2}-7T^{3}+49T^{4}$$D_4 111+2T+4T^{2}+22T^{3}+121T^{4}$$D_4$
13$1-3T+7T^{2}-39T^{3}+169T^{4}$$D_4 171+4T+28T^{2}+68T^{3}+289T^{4}$$D_4$
19$1+T-22T^{2}+19T^{3}+361T^{4}$$D_4 231-3T+22T^{2}-69T^{3}+529T^{4}$$D_4$
29$1+T+13T^{2}+29T^{3}+841T^{4}$$D_4 311+10T+68T^{2}+310T^{3}+961T^{4}$$D_4$
37$(1-8T+37T^{2})(1+4T+37T^{2})$$C_2$$\times$$C_2 411-7T+37T^{2}-287T^{3}+1681T^{4}$$D_4$
43$1-4T+36T^{2}-172T^{3}+1849T^{4}$$D_4 47(1-4T+47T^{2})(1+12T+47T^{2})$$C_2$$\times$$C_2$
53$1-14T+136T^{2}-742T^{3}+2809T^{4}$$D_4 591-T+43T^{2}-59T^{3}+3481T^{4}$$D_4$
61$(1-12T+61T^{2})(1+10T+61T^{2})$$C_2 671-5T-10T^{2}-335T^{3}+4489T^{4}$$D_4$
71$1+58T^{2}+5041T^{4}$$V_4 731+3T+28T^{2}+219T^{3}+5329T^{4}$$D_4$
79$(1+10T+79T^{2})^{2}$$C_2 831-2T+58T^{2}-166T^{3}+6889T^{4}$$D_4$
89$1-3T-32T^{2}-267T^{3}+7921T^{4}$$D_4 97(1-8T+97T^{2})(1+14T+97T^{2})$$C_2$$\times$$C_2$
$$$L(s,A) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$$