Dirichlet series
| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s − 4·11-s − 2·15-s + 2·17-s + 12·19-s − 4·21-s + 8·23-s − 6·25-s + 27-s + 6·29-s − 12·31-s − 4·33-s + 8·35-s − 4·37-s − 6·41-s + 4·43-s − 2·45-s + 2·49-s + 2·51-s − 2·53-s + 8·55-s + 12·57-s − 4·59-s + 12·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.516·15-s + 0.485·17-s + 2.75·19-s − 0.872·21-s + 1.66·23-s − 6/5·25-s + 0.192·27-s + 1.11·29-s − 2.15·31-s − 0.696·33-s + 1.35·35-s − 0.657·37-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 2/7·49-s + 0.280·51-s − 0.274·53-s + 1.07·55-s + 1.58·57-s − 0.520·59-s + 1.53·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1728\) = \(2^{6} \cdot 3^{3}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.110178\) |
| Root analytic conductor: | \(0.576135\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1728,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5909322453\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5909322453\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 3 | $C_1$ | \( 1 - T \) | ||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.c_k |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.e_o | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.e_w | |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.13.a_w | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.17.ac_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.19.am_cs | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.23.ai_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.29.ag_cg | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.m_dq | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.e_o | |
| 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$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.53.c_ec | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.59.e_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.61.am_dq | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.67.m_cs | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.i_fm | |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.a_bu | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.79.ae_ew | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.83.ae_gk | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.g_gw | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.97.aq_io | |
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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
−19.0038467807, −18.8585099876, −18.0808264901, −17.6873258241, −16.5594222959, −16.2503860035, −15.6969681316, −15.5766594288, −14.7395675207, −13.9963410512, −13.4092975284, −13.0105598262, −12.3221586340, −11.7743766738, −11.0205906871, −10.1744110310, −9.65679775515, −9.11342494550, −8.17328234983, −7.26958585489, −7.26646731082, −5.80268955255, −5.01357758336, −3.44334336791, −3.14826068230, 3.14826068230, 3.44334336791, 5.01357758336, 5.80268955255, 7.26646731082, 7.26958585489, 8.17328234983, 9.11342494550, 9.65679775515, 10.1744110310, 11.0205906871, 11.7743766738, 12.3221586340, 13.0105598262, 13.4092975284, 13.9963410512, 14.7395675207, 15.5766594288, 15.6969681316, 16.2503860035, 16.5594222959, 17.6873258241, 18.0808264901, 18.8585099876, 19.0038467807