Dirichlet series
| L(s) = 1 | − 2-s − 4-s − 4·5-s − 5·7-s + 8-s + 4·10-s − 4·11-s − 2·13-s + 5·14-s + 3·16-s − 3·17-s − 19-s + 4·20-s + 4·22-s − 11·23-s + 6·25-s + 2·26-s + 5·28-s + 2·31-s − 3·32-s + 3·34-s + 20·35-s − 6·37-s + 38-s − 4·40-s − 41-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.88·7-s + 0.353·8-s + 1.26·10-s − 1.20·11-s − 0.554·13-s + 1.33·14-s + 3/4·16-s − 0.727·17-s − 0.229·19-s + 0.894·20-s + 0.852·22-s − 2.29·23-s + 6/5·25-s + 0.392·26-s + 0.944·28-s + 0.359·31-s − 0.530·32-s + 0.514·34-s + 3.38·35-s − 0.986·37-s + 0.162·38-s − 0.632·40-s − 0.156·41-s + 0.603·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 47142 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47142 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(47142\) = \(2 \cdot 3^{5} \cdot 97\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.00581\) |
| Root analytic conductor: | \(1.31671\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 47142,\ (\ :1/2, 1/2),\ 1)\) |
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | |
| 3 | \( 1 \) | |||
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) | ||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.7.f_p | |
| 11 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.11.e_n | |
| 13 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_j | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.19.b_l | |
| 23 | $D_{4}$ | \( 1 + 11 T + 73 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.23.l_cv | |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) | 2.29.a_abm | |
| 31 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.31.ac_f | |
| 37 | $D_{4}$ | \( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.37.g_cv | |
| 41 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.41.b_ac | |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.43.a_w | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.47.aj_cg | |
| 53 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.53.d_br | |
| 59 | $D_{4}$ | \( 1 + 15 T + 145 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.59.p_fp | |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.a_w | |
| 67 | $D_{4}$ | \( 1 + 7 T + 83 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.67.h_df | |
| 71 | $D_{4}$ | \( 1 + 3 T - 35 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.71.d_abj | |
| 73 | $D_{4}$ | \( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} \) | 2.73.an_dj | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.79.ab_bw | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.83.ao_hi | |
| 89 | $D_{4}$ | \( 1 + 5 T - 41 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.89.f_abp | |
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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
−15.3899496779, −15.2399875056, −14.2964627625, −13.8268997217, −13.4158627469, −12.8764492033, −12.3162596957, −12.2212276657, −11.7946210129, −10.8024358161, −10.6571443910, −9.92764923173, −9.74715590311, −9.11517287816, −8.55452173871, −8.10474671632, −7.61059765415, −7.33273786488, −6.39239505635, −6.06959117464, −5.13874561000, −4.39184627481, −3.78810662347, −3.31968728123, −2.43276960238, 0, 0, 2.43276960238, 3.31968728123, 3.78810662347, 4.39184627481, 5.13874561000, 6.06959117464, 6.39239505635, 7.33273786488, 7.61059765415, 8.10474671632, 8.55452173871, 9.11517287816, 9.74715590311, 9.92764923173, 10.6571443910, 10.8024358161, 11.7946210129, 12.2212276657, 12.3162596957, 12.8764492033, 13.4158627469, 13.8268997217, 14.2964627625, 15.2399875056, 15.3899496779