Dirichlet series
| L(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) = 1 | − 1.41·2-s − 0.577·3-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1.41·8-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 + 3/5·25-s − 1.17·26-s − 0.384·27-s − 0.185·29-s − 0.365·30-s − 1.79·31-s + 0.348·33-s + 1.37·34-s − 0.169·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(277\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0176617\) |
| Root analytic conductor: | \(0.364551\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 277,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1431366605\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1431366605\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 277 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) | |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.2.c_e |
| 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_b | |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.b_ac | |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.7.ab_d | |
| 11 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.11.c_e | |
| 13 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.13.ad_h | |
| 17 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.17.e_bc | |
| 19 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.19.b_aw | |
| 23 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.23.ad_w | |
| 29 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.29.b_n | |
| 31 | $D_{4}$ | \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.31.k_cq | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.37.ae_bq | |
| 41 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.41.ah_bl | |
| 43 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.43.ae_bk | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.47.i_bu | |
| 53 | $D_{4}$ | \( 1 - 14 T + 136 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.53.ao_fg | |
| 59 | $D_{4}$ | \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.59.ab_br | |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.ac_c | |
| 67 | $D_{4}$ | \( 1 - 5 T - 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.67.af_ak | |
| 71 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) | 2.71.a_cg | |
| 73 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.73.d_bc | |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | 2.79.u_jy | |
| 83 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.83.ac_cg | |
| 89 | $D_{4}$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.89.ad_abg | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.97.g_de | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
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