Dirichlet series
| L(s) = 1 | + 2-s + 4-s − 2·5-s + 2·7-s + 8-s − 2·10-s − 2·13-s + 2·14-s + 16-s + 2·17-s + 2·19-s − 2·20-s + 6·23-s − 2·26-s + 2·28-s − 2·29-s + 5·31-s + 32-s + 2·34-s − 4·35-s − 4·37-s + 2·38-s − 2·40-s − 5·41-s + 2·43-s + 6·46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s + 0.353·8-s − 0.632·10-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s − 0.392·26-s + 0.377·28-s − 0.371·29-s + 0.898·31-s + 0.176·32-s + 0.342·34-s − 0.676·35-s − 0.657·37-s + 0.324·38-s − 0.316·40-s − 0.780·41-s + 0.304·43-s + 0.884·46-s + 0.875·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(373248\) = \(2^{9} \cdot 3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(23.7986\) |
| Root analytic conductor: | \(2.20870\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 373248,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.706567001\) |
| \(L(\frac12)\) | \(\approx\) | \(2.706567001\) |
| \(L(\frac{3}{2})\) | 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 | $C_1$ | \( 1 - T \) | |
| 3 | \( 1 \) | |||
| good | 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_e |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.7.ac_c | |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.11.a_ac | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.c_s | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.ac_ao | |
| 19 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.19.ac_ae | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.23.ag_t | |
| 29 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_e | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.af_ck | |
| 37 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.37.e_bm | |
| 41 | $D_{4}$ | \( 1 + 5 T + 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.41.f_e | |
| 43 | $D_{4}$ | \( 1 - 2 T - 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.43.ac_aq | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.ag_dq | |
| 53 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_cs | |
| 59 | $D_{4}$ | \( 1 - 6 T + 106 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.59.ag_ec | |
| 61 | $D_{4}$ | \( 1 - 2 T - 88 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.61.ac_adk | |
| 67 | $D_{4}$ | \( 1 - 14 T + 128 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.67.ao_ey | |
| 71 | $D_{4}$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.71.d_aby | |
| 73 | $D_{4}$ | \( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.73.ac_fe | |
| 79 | $D_{4}$ | \( 1 - 14 T + 155 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.79.ao_fz | |
| 83 | $D_{4}$ | \( 1 + 6 T + 136 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.83.g_fg | |
| 89 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.89.e_cv | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.97.e_cz | |
| show more | ||||
| show less | ||||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.7412305948, −12.4834068543, −12.0665737071, −11.6769146090, −11.4057718374, −10.9771969804, −10.5597725541, −10.0968574986, −9.56678592682, −9.14404043651, −8.49208927116, −8.13016381145, −7.78460344950, −7.23156284254, −6.92923448743, −6.45232359612, −5.59054556232, −5.29204066357, −4.89331241160, −4.26057010935, −3.83871355389, −3.20119760311, −2.64828856279, −1.82201570873, −0.858175811997, 0.858175811997, 1.82201570873, 2.64828856279, 3.20119760311, 3.83871355389, 4.26057010935, 4.89331241160, 5.29204066357, 5.59054556232, 6.45232359612, 6.92923448743, 7.23156284254, 7.78460344950, 8.13016381145, 8.49208927116, 9.14404043651, 9.56678592682, 10.0968574986, 10.5597725541, 10.9771969804, 11.4057718374, 11.6769146090, 12.0665737071, 12.4834068543, 12.7412305948