Dirichlet series
| L(s) = 1 | + 2-s − 3-s + 4-s − 5·5-s − 6-s + 8-s − 4·9-s − 5·10-s + 4·11-s − 12-s − 6·13-s + 5·15-s + 16-s − 7·17-s − 4·18-s − 5·20-s + 4·22-s + 2·23-s − 24-s + 11·25-s − 6·26-s + 6·27-s + 3·29-s + 5·30-s + 31-s + 32-s − 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 2.23·5-s − 0.408·6-s + 0.353·8-s − 4/3·9-s − 1.58·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s + 1.29·15-s + 1/4·16-s − 1.69·17-s − 0.942·18-s − 1.11·20-s + 0.852·22-s + 0.417·23-s − 0.204·24-s + 11/5·25-s − 1.17·26-s + 1.15·27-s + 0.557·29-s + 0.912·30-s + 0.179·31-s + 0.176·32-s − 0.696·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 14992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(14992\) = \(2^{4} \cdot 937\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.955902\) |
| Root analytic conductor: | \(0.988788\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 14992,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 - T \) | |
| 937 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 53 T + p T^{2} ) \) | ||
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.3.b_f |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.f_o | |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) | 2.7.a_a | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.11.ae_k | |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.13.g_s | |
| 17 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.17.h_bk | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_q | |
| 29 | $C_4$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.29.ad_bf | |
| 31 | $D_{4}$ | \( 1 - T - 13 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.31.ab_an | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.e_o | |
| 41 | $D_{4}$ | \( 1 + 3 T - 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_aq | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.43.c_g | |
| 47 | $D_{4}$ | \( 1 - T + 70 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.47.ab_cs | |
| 53 | $D_{4}$ | \( 1 - T - 15 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.53.ab_ap | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.59.c_cs | |
| 61 | $D_{4}$ | \( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.61.ab_bs | |
| 67 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.67.g_dk | |
| 71 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.71.af_y | |
| 73 | $D_{4}$ | \( 1 + 7 T + 127 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.73.h_ex | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.79.ak_dy | |
| 83 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.83.ag_cw | |
| 89 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) | 2.89.a_afe | |
| 97 | $D_{4}$ | \( 1 + 23 T + 301 T^{2} + 23 p T^{3} + p^{2} T^{4} \) | 2.97.x_lp | |
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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
−16.3190809692, −15.6960408096, −15.3153669107, −14.8969785961, −14.6360024168, −13.9423655191, −13.5488633752, −12.5564915697, −12.2578949841, −11.9034855504, −11.4475673851, −11.2786084058, −10.7036485244, −9.89118982662, −8.88665554902, −8.72080749416, −7.89925451367, −7.40401517730, −6.72796819188, −6.38240357587, −5.28142824728, −4.72321870941, −4.16652832534, −3.44192326026, −2.55452430678, 0, 2.55452430678, 3.44192326026, 4.16652832534, 4.72321870941, 5.28142824728, 6.38240357587, 6.72796819188, 7.40401517730, 7.89925451367, 8.72080749416, 8.88665554902, 9.89118982662, 10.7036485244, 11.2786084058, 11.4475673851, 11.9034855504, 12.2578949841, 12.5564915697, 13.5488633752, 13.9423655191, 14.6360024168, 14.8969785961, 15.3153669107, 15.6960408096, 16.3190809692