Dirichlet series
| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s + 9-s − 13-s − 4·14-s − 16-s + 4·17-s + 18-s + 4·19-s + 8·23-s − 6·25-s − 26-s + 4·28-s − 4·29-s − 4·31-s + 5·32-s + 4·34-s − 36-s + 4·37-s + 4·38-s − 16·43-s + 8·46-s + 2·49-s − 6·50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.277·13-s − 1.06·14-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.917·19-s + 1.66·23-s − 6/5·25-s − 0.196·26-s + 0.755·28-s − 0.742·29-s − 0.718·31-s + 0.883·32-s + 0.685·34-s − 1/6·36-s + 0.657·37-s + 0.648·38-s − 2.43·43-s + 1.17·46-s + 2/7·49-s − 0.848·50-s + 0.138·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1872\) = \(2^{4} \cdot 3^{2} \cdot 13\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.119360\) |
| Root analytic conductor: | \(0.587780\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1872,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6967946687\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6967946687\) |
| \(L(\frac{3}{2})\) | 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 | $C_2$ | \( 1 - T + p T^{2} \) | |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | ||
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.a_g |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.e_o | |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.a_g | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.17.ae_bm | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.19.ae_bm | |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.29.e_ac | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.e_be | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.ae_ck | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.a_bu | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.43.q_fe | |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.47.a_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.53.ae_dq | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.59.ai_cs | |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.61.e_ew | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.67.e_dy | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.i_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.73.am_gk | |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.79.aq_io | |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.83.ai_ha | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.i_hi | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.97.am_ig | |
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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, −18.4282714936, −18.0808264901, −17.2849148009, −16.5594222959, −16.4135833599, −15.5766594288, −14.9380436528, −14.7395675207, −13.6745032861, −13.4092975284, −12.9568449998, −12.3221586340, −11.8538765065, −11.0205906871, −9.95280234664, −9.65679775515, −9.19274055467, −8.17328234983, −7.26958585489, −6.54556545043, −5.65486780280, −5.01357758336, −3.71205447024, −3.14826068230, 3.14826068230, 3.71205447024, 5.01357758336, 5.65486780280, 6.54556545043, 7.26958585489, 8.17328234983, 9.19274055467, 9.65679775515, 9.95280234664, 11.0205906871, 11.8538765065, 12.3221586340, 12.9568449998, 13.4092975284, 13.6745032861, 14.7395675207, 14.9380436528, 15.5766594288, 16.4135833599, 16.5594222959, 17.2849148009, 18.0808264901, 18.4282714936, 19.0038467807