Dirichlet series
| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 5·7-s + 8-s + 9-s + 10-s − 12-s + 7·13-s + 5·14-s − 15-s + 3·16-s + 6·17-s − 18-s − 11·19-s + 20-s − 5·21-s + 24-s − 4·25-s − 7·26-s + 27-s + 5·28-s − 6·29-s + 30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.94·13-s + 1.33·14-s − 0.258·15-s + 3/4·16-s + 1.45·17-s − 0.235·18-s − 2.52·19-s + 0.223·20-s − 1.09·21-s + 0.204·24-s − 4/5·25-s − 1.37·26-s + 0.192·27-s + 0.944·28-s − 1.11·29-s + 0.182·30-s + 0.718·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(810\) = \(2 \cdot 3^{4} \cdot 5\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0516463\) |
| Root analytic conductor: | \(0.476716\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 810,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3289823157\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3289823157\) |
| \(L(\frac{3}{2})\) | 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 | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | |
| 3 | $C_1$ | \( 1 - T \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.f_s |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.11.a_w | |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.13.ah_bk | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.17.ag_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.19.l_co | |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.23.a_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.g_cg | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.31.ae_be | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.37.an_ds | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.g_de | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.ae_cc | |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.47.a_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.g_ec | |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.59.a_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.l_fc | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.ab_ek | |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.71.a_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.73.f_fc | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) | 2.79.az_li | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.83.am_gk | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) | 2.89.as_gw | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) | 2.97.r_ga | |
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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
−19.5076338893, −19.1626037497, −18.6983747433, −18.5592411054, −17.5495821026, −16.9011141517, −16.3000171353, −16.0225385141, −14.9691010860, −14.8921107674, −13.5821136983, −13.3351916393, −12.7156394907, −12.1436052702, −10.9087282930, −10.3916050944, −9.42919920821, −9.30558712287, −8.21765036746, −7.94123132226, −6.42217617653, −6.04893540001, −4.04304401380, −3.36585804146, 3.36585804146, 4.04304401380, 6.04893540001, 6.42217617653, 7.94123132226, 8.21765036746, 9.30558712287, 9.42919920821, 10.3916050944, 10.9087282930, 12.1436052702, 12.7156394907, 13.3351916393, 13.5821136983, 14.8921107674, 14.9691010860, 16.0225385141, 16.3000171353, 16.9011141517, 17.5495821026, 18.5592411054, 18.6983747433, 19.1626037497, 19.5076338893