Dirichlet series
| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 4·5-s + 6·6-s − 7-s − 3·8-s + 9-s + 12·10-s − 8·12-s − 13-s + 3·14-s + 8·15-s + 3·16-s − 3·18-s − 4·19-s − 16·20-s + 2·21-s + 2·23-s + 6·24-s + 5·25-s + 3·26-s − 2·27-s − 4·28-s − 24·30-s + 4·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 1.78·5-s + 2.44·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 3.79·10-s − 2.30·12-s − 0.277·13-s + 0.801·14-s + 2.06·15-s + 3/4·16-s − 0.707·18-s − 0.917·19-s − 3.57·20-s + 0.436·21-s + 0.417·23-s + 1.22·24-s + 25-s + 0.588·26-s − 0.384·27-s − 0.755·28-s − 4.38·30-s + 0.718·31-s − 1.06·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 893 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(893\) = \(19 \cdot 47\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.0569384\) |
| Root analytic conductor: | \(0.488485\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 893,\ (\ :1/2, 1/2),\ -1)\) |
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 | 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) | |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) | ||
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.2.d_f |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) | 2.3.c_d | |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.5.e_l | |
| 7 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_h | |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) | 2.11.a_t | |
| 13 | $D_{4}$ | \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \) | 2.13.b_an | |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.17.a_s | |
| 23 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_ah | |
| 29 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) | 2.29.a_abv | |
| 31 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.31.ae_m | |
| 37 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.37.o_ee | |
| 41 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.41.ab_bh | |
| 43 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) | 2.43.a_acd | |
| 53 | $D_{4}$ | \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.53.g_k | |
| 59 | $D_{4}$ | \( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.59.ak_ec | |
| 61 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.61.ag_cs | |
| 67 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.67.ae_co | |
| 71 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.71.ae_as | |
| 73 | $D_{4}$ | \( 1 - 14 T + 177 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.73.ao_gv | |
| 79 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) | 2.79.a_ga | |
| 83 | $D_{4}$ | \( 1 + 3 T - 49 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.83.d_abx | |
| 89 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.89.c_m | |
| 97 | $D_{4}$ | \( 1 - 2 T - 121 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.97.ac_aer | |
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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
−19.4901123623, −19.1188559049, −18.9941196049, −18.1962238572, −17.5803652188, −17.1605094556, −16.8851484436, −16.0601464230, −15.7560683073, −15.1661355976, −14.3513730910, −13.1074126825, −12.3110918435, −11.8701368771, −11.2395985780, −10.7353474786, −10.0294591025, −9.31948512140, −8.38401905615, −8.11597358074, −7.20784764151, −6.47935710087, −5.14407534839, −3.75241654741, 0, 3.75241654741, 5.14407534839, 6.47935710087, 7.20784764151, 8.11597358074, 8.38401905615, 9.31948512140, 10.0294591025, 10.7353474786, 11.2395985780, 11.8701368771, 12.3110918435, 13.1074126825, 14.3513730910, 15.1661355976, 15.7560683073, 16.0601464230, 16.8851484436, 17.1605094556, 17.5803652188, 18.1962238572, 18.9941196049, 19.1188559049, 19.4901123623