Dirichlet series
| L(s) = 1 | − 2-s + 3-s − 2·4-s − 2·5-s − 6-s − 7-s + 3·8-s + 9-s + 2·10-s + 4·11-s − 2·12-s − 5·13-s + 14-s − 2·15-s + 16-s − 18-s − 3·19-s + 4·20-s − 21-s − 4·22-s − 5·23-s + 3·24-s + 2·25-s + 5·26-s + 27-s + 2·28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s − 0.577·12-s − 1.38·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.688·19-s + 0.894·20-s − 0.218·21-s − 0.852·22-s − 1.04·23-s + 0.612·24-s + 2/5·25-s + 0.980·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 18981 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18981 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(18981\) = \(3^{3} \cdot 19 \cdot 37\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.21024\) |
| Root analytic conductor: | \(1.04886\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 18981,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 - T \) | |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.2.b_d |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.c_c | |
| 7 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_l | |
| 11 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.11.ae_u | |
| 13 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.13.f_p | |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) | 2.17.a_ae | |
| 23 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.23.f_bk | |
| 29 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_bq | |
| 31 | $D_{4}$ | \( 1 + 11 T + 75 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.31.l_cx | |
| 41 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.41.ad_bo | |
| 43 | $D_{4}$ | \( 1 - 3 T - 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.43.ad_abe | |
| 47 | $D_{4}$ | \( 1 + 2 T - 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.47.c_abp | |
| 53 | $D_{4}$ | \( 1 + 8 T + 79 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.53.i_db | |
| 59 | $D_{4}$ | \( 1 - 7 T + 89 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.59.ah_dl | |
| 61 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) | 2.61.a_n | |
| 67 | $D_{4}$ | \( 1 + 9 T + 80 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.67.j_dc | |
| 71 | $D_{4}$ | \( 1 + 2 T + 100 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.71.c_dw | |
| 73 | $D_{4}$ | \( 1 - 4 T - 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.73.ae_ap | |
| 79 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.79.ab_d | |
| 83 | $D_{4}$ | \( 1 + 3 T - 57 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.83.d_acf | |
| 89 | $D_{4}$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.89.ad_bf | |
| 97 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) | 2.97.a_acp | |
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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
−16.1482997533, −15.6368821880, −14.8440890298, −14.6651930845, −14.2895266720, −13.7943626018, −13.0611189594, −12.7180857677, −12.2591889304, −11.7336836344, −11.1070240393, −10.4802258698, −9.73757710175, −9.58323117885, −9.00467896753, −8.61829303666, −7.97634358103, −7.51564039708, −6.96683425817, −6.20453593355, −5.20665942770, −4.44962768490, −3.98691484258, −3.30752797181, −1.91408666502, 0, 1.91408666502, 3.30752797181, 3.98691484258, 4.44962768490, 5.20665942770, 6.20453593355, 6.96683425817, 7.51564039708, 7.97634358103, 8.61829303666, 9.00467896753, 9.58323117885, 9.73757710175, 10.4802258698, 11.1070240393, 11.7336836344, 12.2591889304, 12.7180857677, 13.0611189594, 13.7943626018, 14.2895266720, 14.6651930845, 14.8440890298, 15.6368821880, 16.1482997533