Dirichlet series
| L(s) = 1 | − 2·2-s + 4-s − 2·5-s − 3·7-s + 9-s + 4·10-s − 4·11-s − 13-s + 6·14-s + 16-s − 3·17-s − 2·18-s − 2·20-s + 8·22-s − 4·23-s − 2·25-s + 2·26-s − 3·28-s + 9·31-s + 2·32-s + 6·34-s + 6·35-s + 36-s − 7·37-s − 41-s − 43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.894·5-s − 1.13·7-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 0.277·13-s + 1.60·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.447·20-s + 1.70·22-s − 0.834·23-s − 2/5·25-s + 0.392·26-s − 0.566·28-s + 1.61·31-s + 0.353·32-s + 1.02·34-s + 1.01·35-s + 1/6·36-s − 1.15·37-s − 0.156·41-s − 0.152·43-s − 0.603·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 3897 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3897 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(3897\) = \(3^{2} \cdot 433\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.248476\) |
| Root analytic conductor: | \(0.706026\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 3897,\ (\ :1/2, 1/2),\ -1)\) |
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 433 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 26 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.2.c_d |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_g | |
| 7 | $D_{4}$ | \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.7.d_g | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.e_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.13.b_u | |
| 17 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.17.d_e | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.23.e_ag | |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.29.a_k | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.aj_ck | |
| 37 | $D_{4}$ | \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.37.h_bw | |
| 41 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.41.b_y | |
| 43 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.43.b_k | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.47.af_ak | |
| 53 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.53.b_m | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.am_eo | |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) | 2.61.a_cs | |
| 67 | $D_{4}$ | \( 1 + 13 T + 138 T^{2} + 13 p T^{3} + p^{2} T^{4} \) | 2.67.n_fi | |
| 71 | $D_{4}$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.71.j_cs | |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) | 2.73.a_cs | |
| 79 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.79.ag_cc | |
| 83 | $D_{4}$ | \( 1 - T - 126 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.83.ab_aew | |
| 89 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.89.ae_dq | |
| 97 | $D_{4}$ | \( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.97.ac_abm | |
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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
−18.0833181124, −17.7308329983, −17.2868630615, −16.5831980740, −16.0224982285, −15.6529230159, −15.4416531115, −14.6775520441, −13.6014305360, −13.4689602557, −12.7102119744, −12.0301072875, −11.7399442665, −10.7035031288, −10.2095682322, −9.89047025611, −9.18176978282, −8.56340932560, −8.04119623905, −7.47757615408, −6.71606663193, −5.92313395768, −4.77902395116, −3.81662957763, −2.67424177483, 0, 2.67424177483, 3.81662957763, 4.77902395116, 5.92313395768, 6.71606663193, 7.47757615408, 8.04119623905, 8.56340932560, 9.18176978282, 9.89047025611, 10.2095682322, 10.7035031288, 11.7399442665, 12.0301072875, 12.7102119744, 13.4689602557, 13.6014305360, 14.6775520441, 15.4416531115, 15.6529230159, 16.0224982285, 16.5831980740, 17.2868630615, 17.7308329983, 18.0833181124