Dirichlet series
| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 2·7-s + 8-s + 3·9-s − 4·11-s − 2·12-s + 8·13-s − 2·14-s + 16-s + 4·17-s + 3·18-s + 4·21-s − 4·22-s + 8·23-s − 2·24-s − 6·25-s + 8·26-s − 4·27-s − 2·28-s + 4·29-s + 32-s + 8·33-s + 4·34-s + 3·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.755·7-s + 0.353·8-s + 9-s − 1.20·11-s − 0.577·12-s + 2.21·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.872·21-s − 0.852·22-s + 1.66·23-s − 0.408·24-s − 6/5·25-s + 1.56·26-s − 0.769·27-s − 0.377·28-s + 0.742·29-s + 0.176·32-s + 1.39·33-s + 0.685·34-s + 1/2·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(28224\) = \(2^{6} \cdot 3^{2} \cdot 7^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.79958\) |
| Root analytic conductor: | \(1.15822\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 28224,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.279824216\) |
| \(L(\frac12)\) | \(\approx\) | \(1.279824216\) |
| \(L(\frac{3}{2})\) | 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 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| 7 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.a_g |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.e_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.13.ai_bm | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.17.ae_bm | |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.a_w | |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.ae_bu | |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.31.a_ck | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.e_o | |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.41.m_eo | |
| 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$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.ai_dq | |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.53.am_fm | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.i_cs | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) | 2.61.aq_ha | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.67.au_hq | |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.a_da | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.73.ae_di | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.79.ai_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.83.q_ig | |
| 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$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.97.u_ks | |
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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
−15.4648968402, −15.2472562838, −14.2531155803, −13.7262953554, −13.4821859369, −12.9353530446, −12.6521323881, −12.1318844803, −11.3731892768, −11.3351214109, −10.5195197796, −10.3639838911, −9.80165984984, −8.96665534829, −8.35691896663, −7.81839476073, −6.84552480603, −6.77602051524, −5.94586279153, −5.33985014788, −5.29886217645, −4.00502420742, −3.62482887082, −2.67859582553, −1.15780346499, 1.15780346499, 2.67859582553, 3.62482887082, 4.00502420742, 5.29886217645, 5.33985014788, 5.94586279153, 6.77602051524, 6.84552480603, 7.81839476073, 8.35691896663, 8.96665534829, 9.80165984984, 10.3639838911, 10.5195197796, 11.3351214109, 11.3731892768, 12.1318844803, 12.6521323881, 12.9353530446, 13.4821859369, 13.7262953554, 14.2531155803, 15.2472562838, 15.4648968402