Properties

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

Related objects

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Normalization:  

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}$$D_4$
11$1+2T+4T^{2}+22T^{3}+121T^{4}$$D_4$
13$1-3T+7T^{2}-39T^{3}+169T^{4}$$D_4$
17$1+4T+28T^{2}+68T^{3}+289T^{4}$$D_4$
19$1+T-22T^{2}+19T^{3}+361T^{4}$$D_4$
23$1-3T+22T^{2}-69T^{3}+529T^{4}$$D_4$
29$1+T+13T^{2}+29T^{3}+841T^{4}$$D_4$
31$1+10T+68T^{2}+310T^{3}+961T^{4}$$D_4$
37$(1-8T+37T^{2})(1+4T+37T^{2})$$C_2$$\times$$C_2$
41$1-7T+37T^{2}-287T^{3}+1681T^{4}$$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}$$D_4$
59$1-T+43T^{2}-59T^{3}+3481T^{4}$$D_4$
61$(1-12T+61T^{2})(1+10T+61T^{2})$$C_2$
67$1-5T-10T^{2}-335T^{3}+4489T^{4}$$D_4$
71$1+58T^{2}+5041T^{4}$$V_4$
73$1+3T+28T^{2}+219T^{3}+5329T^{4}$$D_4$
79$(1+10T+79T^{2})^{2}$$C_2$
83$1-2T+58T^{2}-166T^{3}+6889T^{4}$$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$
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\[\begin{equation} L(s,A) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{equation}\]

Imaginary part of the first few zeros on the critical line

−19.8383227474, −19.1745567864, −18.6003292006, −18.0126226421, −17.8685107407, −17.0212695469, −16.5112504108, −15.8575391568, −15.0261185798, −14.2430115276, −13.1187710094, −12.9354901068, −11.4860616488, −11.0355703521, −10.3714949973, −9.31193442615, −8.74527851838, −8.01150574486, −7.07871685645, −5.58349348623, −4.30532032866, 4.30532032866, 5.58349348623, 7.07871685645, 8.01150574486, 8.74527851838, 9.31193442615, 10.3714949973, 11.0355703521, 11.4860616488, 12.9354901068, 13.1187710094, 14.2430115276, 15.0261185798, 15.8575391568, 16.5112504108, 17.0212695469, 17.8685107407, 18.0126226421, 18.6003292006, 19.1745567864, 19.8383227474

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