Dirichlet series
| L(s) = 1 | − 2·2-s − 3·3-s − 6·5-s + 6·6-s − 2·7-s + 4·8-s + 4·9-s + 12·10-s − 2·11-s − 2·13-s + 4·14-s + 18·15-s − 4·16-s − 6·17-s − 8·18-s + 6·21-s + 4·22-s − 2·23-s − 12·24-s + 18·25-s + 4·26-s − 6·29-s − 36·30-s − 6·31-s + 6·33-s + 12·34-s + 12·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s − 2.68·5-s + 2.44·6-s − 0.755·7-s + 1.41·8-s + 4/3·9-s + 3.79·10-s − 0.603·11-s − 0.554·13-s + 1.06·14-s + 4.64·15-s − 16-s − 1.45·17-s − 1.88·18-s + 1.30·21-s + 0.852·22-s − 0.417·23-s − 2.44·24-s + 18/5·25-s + 0.784·26-s − 1.11·29-s − 6.57·30-s − 1.07·31-s + 1.04·33-s + 2.05·34-s + 2.02·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 5547 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5547 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(5547\) = \(3 \cdot 43^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.353681\) |
| Root analytic conductor: | \(0.771175\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 5547,\ (\ :1/2, 1/2),\ 1)\) |
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | |
| 43 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.2.c_e |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.g_s | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.7.c_o | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.c_h | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.13.c_l | |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) | 2.17.g_br | |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.19.a_bi | |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) | 2.23.c_bv | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.g_cg | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.31.g_cp | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.37.ai_cw | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.41.c_bv | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.e_ck | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.53.c_dn | |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.a_aba | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.61.g_ec | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) | 2.67.s_gx | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.71.m_ek | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.73.ao_go | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.79.y_la | |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) | 2.83.abe_pb | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.89.ag_fi | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) | 2.97.as_kl | |
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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
−17.8729368019, −17.4780977310, −16.8238626428, −16.4464822737, −15.9802273957, −15.8966001066, −14.9484426154, −14.7879696813, −13.3423691023, −13.0803935079, −12.3834323879, −11.8360583747, −11.5135334923, −10.7971801876, −10.7500404360, −9.76175813922, −9.20679535768, −8.51053708633, −7.86442320164, −7.51672561269, −6.82871744579, −5.96338923145, −4.74387622428, −4.49472027398, −3.47776319336, 0, 0, 3.47776319336, 4.49472027398, 4.74387622428, 5.96338923145, 6.82871744579, 7.51672561269, 7.86442320164, 8.51053708633, 9.20679535768, 9.76175813922, 10.7500404360, 10.7971801876, 11.5135334923, 11.8360583747, 12.3834323879, 13.0803935079, 13.3423691023, 14.7879696813, 14.9484426154, 15.8966001066, 15.9802273957, 16.4464822737, 16.8238626428, 17.4780977310, 17.8729368019