Dirichlet series
| L(s) = 1 | + 2-s − 2·4-s + 5-s − 4·7-s − 3·8-s − 3·9-s + 10-s + 4·11-s − 4·14-s + 16-s − 4·17-s − 3·18-s − 7·19-s − 2·20-s + 4·22-s − 2·23-s − 2·25-s + 8·28-s + 5·29-s − 3·31-s + 2·32-s − 4·34-s − 4·35-s + 6·36-s − 3·37-s − 7·38-s − 3·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s + 0.447·5-s − 1.51·7-s − 1.06·8-s − 9-s + 0.316·10-s + 1.20·11-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 1.60·19-s − 0.447·20-s + 0.852·22-s − 0.417·23-s − 2/5·25-s + 1.51·28-s + 0.928·29-s − 0.538·31-s + 0.353·32-s − 0.685·34-s − 0.676·35-s + 36-s − 0.493·37-s − 1.13·38-s − 0.474·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(29241\) = \(3^{4} \cdot 19^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.86443\) |
| Root analytic conductor: | \(1.16852\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 29241,\ (\ :1/2, 1/2),\ -1)\) |
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 | 3 | $C_2$ | \( 1 + p T^{2} \) | |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) | ||
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.2.ab_d |
| 5 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.5.ab_d | |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.7.e_j | |
| 11 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.11.ae_u | |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) | 2.13.a_f | |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.17.e_be | |
| 23 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.23.c_ai | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.29.af_ca | |
| 31 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.31.d_bd | |
| 37 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.37.d_l | |
| 41 | $D_{4}$ | \( 1 - 6 T + 4 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.41.ag_e | |
| 43 | $D_{4}$ | \( 1 + 7 T + p T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.43.h_br | |
| 47 | $D_{4}$ | \( 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.47.c_ce | |
| 53 | $D_{4}$ | \( 1 - T + 67 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.53.ab_cp | |
| 59 | $D_{4}$ | \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.59.ah_bt | |
| 61 | $C_4$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.61.n_dt | |
| 67 | $D_{4}$ | \( 1 + 11 T + 53 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.67.l_cb | |
| 71 | $D_{4}$ | \( 1 - 9 T + 159 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.71.aj_gd | |
| 73 | $D_{4}$ | \( 1 + 3 T + 132 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.73.d_fc | |
| 79 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.79.d_d | |
| 83 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) | 2.83.al_el | |
| 89 | $D_{4}$ | \( 1 + 6 T - 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.89.g_ak | |
| 97 | $D_{4}$ | \( 1 - 3 T + 75 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.97.ad_cx | |
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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
−15.3091036474, −14.9854554287, −14.4696214086, −14.0562763698, −13.6567113418, −13.3080960208, −12.8953309480, −12.3482051861, −12.1055434972, −11.2945929240, −10.8693220692, −10.0240906316, −9.81939802331, −9.04139383386, −8.83006148899, −8.54245615071, −7.49680460220, −6.56849086829, −6.29433334377, −5.95135528989, −5.04880745256, −4.40031893183, −3.81453991448, −3.21928800927, −2.19898476262, 0, 2.19898476262, 3.21928800927, 3.81453991448, 4.40031893183, 5.04880745256, 5.95135528989, 6.29433334377, 6.56849086829, 7.49680460220, 8.54245615071, 8.83006148899, 9.04139383386, 9.81939802331, 10.0240906316, 10.8693220692, 11.2945929240, 12.1055434972, 12.3482051861, 12.8953309480, 13.3080960208, 13.6567113418, 14.0562763698, 14.4696214086, 14.9854554287, 15.3091036474