Dirichlet series
| L(s) = 1 | − 2-s − 4·3-s − 4-s − 4·5-s + 4·6-s − 8·7-s + 3·8-s + 7·9-s + 4·10-s + 4·12-s − 6·13-s + 8·14-s + 16·15-s − 16-s − 6·17-s − 7·18-s − 12·19-s + 4·20-s + 32·21-s + 4·23-s − 12·24-s + 6·25-s + 6·26-s − 4·27-s + 8·28-s − 14·29-s − 16·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 1/2·4-s − 1.78·5-s + 1.63·6-s − 3.02·7-s + 1.06·8-s + 7/3·9-s + 1.26·10-s + 1.15·12-s − 1.66·13-s + 2.13·14-s + 4.13·15-s − 1/4·16-s − 1.45·17-s − 1.64·18-s − 2.75·19-s + 0.894·20-s + 6.98·21-s + 0.834·23-s − 2.44·24-s + 6/5·25-s + 1.17·26-s − 0.769·27-s + 1.51·28-s − 2.59·29-s − 2.92·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 89888 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89888 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(89888\) = \(2^{5} \cdot 53^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(5.73133\) |
| Root analytic conductor: | \(1.54726\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(3\) |
| Selberg data: | \((4,\ 89888,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 53 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.e_j |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k | |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.7.i_be | |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.11.a_w | |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) | 2.13.g_bj | |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) | 2.17.g_br | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.19.m_cv | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.23.ae_z | |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) | 2.29.o_ed | |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.31.ai_da | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) | 2.37.g_t | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.a_bu | |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.43.a_de | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.47.ae_de | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.59.am_dm | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.61.u_ik | |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) | 2.67.y_ks | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.71.e_fh | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.73.ae_ek | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.79.ae_fx | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.83.ae_gf | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.89.e_bm | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.97.g_hf | |
| 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
−15.2511670091, −14.6520461766, −13.5299185736, −13.2192960322, −12.7547962161, −12.6214261523, −12.0981046476, −11.8169228658, −11.1708289929, −10.7826393650, −10.3965368022, −10.1489334004, −9.26423882619, −9.16916541994, −8.58472990070, −7.70052891010, −7.20736947999, −6.83243855912, −6.35138688453, −6.04347889941, −5.20643941434, −4.50862835068, −4.19896819700, −3.53898731695, −2.53068466458, 0, 0, 0, 2.53068466458, 3.53898731695, 4.19896819700, 4.50862835068, 5.20643941434, 6.04347889941, 6.35138688453, 6.83243855912, 7.20736947999, 7.70052891010, 8.58472990070, 9.16916541994, 9.26423882619, 10.1489334004, 10.3965368022, 10.7826393650, 11.1708289929, 11.8169228658, 12.0981046476, 12.6214261523, 12.7547962161, 13.2192960322, 13.5299185736, 14.6520461766, 15.2511670091