Dirichlet series
| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 4·7-s − 3·8-s + 9-s + 2·10-s + 12-s − 3·13-s − 4·14-s − 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s − 2·20-s + 4·21-s + 12·23-s − 3·24-s + 2·25-s + 3·26-s + 27-s + 4·28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.832·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.872·21-s + 2.50·23-s − 0.612·24-s + 2/5·25-s + 0.588·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.365·30-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: | \(0\) |
| Selberg data: | \((4,\ 29241,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.041519614\) |
| \(L(\frac12)\) | \(\approx\) | \(1.041519614\) |
| \(L(\frac{3}{2})\) | 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 | $C_1$ | \( 1 - T \) | |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.2.b_a |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.c_c | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.7.ae_o | |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.11.a_ac | |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.13.d_m | |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.17.c_ag | |
| 23 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.23.am_cs | |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.29.ag_bi | |
| 31 | $D_{4}$ | \( 1 + T + 46 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.31.b_bu | |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.37.ah_bw | |
| 41 | $D_{4}$ | \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_ao | |
| 43 | $D_{4}$ | \( 1 - 7 T + 74 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.43.ah_cw | |
| 47 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.47.ae_ak | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.ag_bi | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.i_cs | |
| 61 | $D_{4}$ | \( 1 + 7 T + 100 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.61.h_dw | |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.67.m_dy | |
| 71 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.71.am_eo | |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.a_bu | |
| 79 | $D_{4}$ | \( 1 + 3 T + 78 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.79.d_da | |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) | 2.83.a_di | |
| 89 | $D_{4}$ | \( 1 - 2 T - 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.89.ac_aew | |
| 97 | $D_{4}$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.97.d_ce | |
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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
−15.1716209388, −14.9334957488, −14.7065250812, −13.9896259477, −13.6099458894, −12.8898473619, −12.2960876342, −11.9649343529, −11.4329217968, −11.0158978449, −10.7390973036, −9.86079047521, −9.28854084856, −8.90824193892, −8.50528525347, −7.79854048569, −7.55316546016, −7.00207154589, −6.35275017136, −5.26257928174, −4.80530411562, −4.17709351755, −3.01541212941, −2.57603016708, −1.17711345864, 1.17711345864, 2.57603016708, 3.01541212941, 4.17709351755, 4.80530411562, 5.26257928174, 6.35275017136, 7.00207154589, 7.55316546016, 7.79854048569, 8.50528525347, 8.90824193892, 9.28854084856, 9.86079047521, 10.7390973036, 11.0158978449, 11.4329217968, 11.9649343529, 12.2960876342, 12.8898473619, 13.6099458894, 13.9896259477, 14.7065250812, 14.9334957488, 15.1716209388