Dirichlet series
| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s − 8·11-s + 12-s − 3·13-s + 2·15-s + 16-s − 2·17-s − 18-s − 14·19-s + 2·20-s + 8·22-s + 8·23-s − 24-s + 2·25-s + 3·26-s + 27-s − 8·29-s − 2·30-s − 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 2.41·11-s + 0.288·12-s − 0.832·13-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 3.21·19-s + 0.447·20-s + 1.70·22-s + 1.66·23-s − 0.204·24-s + 2/5·25-s + 0.588·26-s + 0.192·27-s − 1.48·29-s − 0.365·30-s − 1.43·31-s − 0.176·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 89856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(89856\) = \(2^{8} \cdot 3^{3} \cdot 13\) |
| Sign: | $-1$ |
| Analytic conductor: | \(5.72929\) |
| Root analytic conductor: | \(1.54712\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 89856,\ (\ :1/2, 1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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 | $C_1$ | \( 1 - T \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.ac_c |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.7.a_k | |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.11.i_bm | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.17.c_bi | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.19.o_di | |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.23.ai_ck | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.29.i_cg | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.31.i_cw | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.37.ai_di | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.41.am_eo | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.43.i_di | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.47.am_ew | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.53.am_ec | |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.59.ai_fe | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.61.e_as | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.g_fm | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.71.e_aby | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.ae_g | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.79.ag_eo | |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.83.a_w | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.89.u_kc | |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.a_dq | |
| 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
−14.6792579084, −13.8852555440, −13.1981160403, −13.0814885133, −12.6736117023, −12.6632052580, −11.4847145732, −10.9063902704, −10.7072544481, −10.4866496825, −9.77369970372, −9.43370759693, −8.74330007195, −8.69573966035, −7.76516545555, −7.69025257464, −6.93707785293, −6.52923673371, −5.59584593533, −5.46781891073, −4.59332442535, −3.99394475360, −2.63009703572, −2.57523735650, −1.91249343135, 0, 1.91249343135, 2.57523735650, 2.63009703572, 3.99394475360, 4.59332442535, 5.46781891073, 5.59584593533, 6.52923673371, 6.93707785293, 7.69025257464, 7.76516545555, 8.69573966035, 8.74330007195, 9.43370759693, 9.77369970372, 10.4866496825, 10.7072544481, 10.9063902704, 11.4847145732, 12.6632052580, 12.6736117023, 13.0814885133, 13.1981160403, 13.8852555440, 14.6792579084