Dirichlet series
| L(s) = 1 | − 2-s − 3·3-s + 4-s − 5-s + 3·6-s + 2·7-s − 8-s + 2·9-s + 10-s − 11-s − 3·12-s + 3·13-s − 2·14-s + 3·15-s + 16-s − 8·17-s − 2·18-s − 20-s − 6·21-s + 22-s + 7·23-s + 3·24-s − 3·26-s + 6·27-s + 2·28-s − 6·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.755·7-s − 0.353·8-s + 2/3·9-s + 0.316·10-s − 0.301·11-s − 0.866·12-s + 0.832·13-s − 0.534·14-s + 0.774·15-s + 1/4·16-s − 1.94·17-s − 0.471·18-s − 0.223·20-s − 1.30·21-s + 0.213·22-s + 1.45·23-s + 0.612·24-s − 0.588·26-s + 1.15·27-s + 0.377·28-s − 1.11·29-s − 0.547·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 14192 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14192 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(14192\) = \(2^{4} \cdot 887\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.904894\) |
| Root analytic conductor: | \(0.975325\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 14192,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 + T \) | |
| 887 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 22 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_h |
| 5 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) | 2.5.b_b | |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.7.ac_k | |
| 11 | $D_{4}$ | \( 1 + T - 3 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.11.b_ad | |
| 13 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.13.ad_g | |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.17.i_bq | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.23.ah_bh | |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.29.g_bi | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.g_bu | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.g_bi | |
| 41 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.41.i_bu | |
| 43 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.43.b_l | |
| 47 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.47.c_ck | |
| 53 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.53.e_ac | |
| 59 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.59.f_be | |
| 61 | $D_{4}$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.61.ad_ae | |
| 67 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.67.ag_be | |
| 71 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.71.ag_cy | |
| 73 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) | 2.73.d_a | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 16 T + p T^{2} ) \) | 2.79.p_fm | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) | 2.83.n_du | |
| 89 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) | 2.89.a_ae | |
| 97 | $D_{4}$ | \( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.97.g_gq | |
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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
−16.5140507871, −16.0673636084, −15.5691735145, −15.2104130011, −14.6524259602, −13.9973446073, −13.2527591852, −12.8758260310, −12.2938994115, −11.5258589284, −11.2105068302, −11.1753245861, −10.7429656423, −10.0137716326, −9.11280982793, −8.68341890639, −8.26162002832, −7.35638277668, −6.81179282946, −6.34735463185, −5.49995879701, −5.14236486074, −4.34661968792, −3.21167913785, −1.74617162298, 0, 1.74617162298, 3.21167913785, 4.34661968792, 5.14236486074, 5.49995879701, 6.34735463185, 6.81179282946, 7.35638277668, 8.26162002832, 8.68341890639, 9.11280982793, 10.0137716326, 10.7429656423, 11.1753245861, 11.2105068302, 11.5258589284, 12.2938994115, 12.8758260310, 13.2527591852, 13.9973446073, 14.6524259602, 15.2104130011, 15.5691735145, 16.0673636084, 16.5140507871