Dirichlet series
| L(s) = 1 | + 2-s − 4-s − 2·5-s + 2·7-s − 8-s − 2·10-s + 4·11-s + 5·13-s + 2·14-s − 16-s − 3·17-s + 4·19-s + 2·20-s + 4·22-s − 23-s − 2·25-s + 5·26-s − 2·28-s + 4·29-s + 7·31-s − 5·32-s − 3·34-s − 4·35-s + 2·37-s + 4·38-s + 2·40-s + 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.755·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 1.38·13-s + 0.534·14-s − 1/4·16-s − 0.727·17-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.208·23-s − 2/5·25-s + 0.980·26-s − 0.377·28-s + 0.742·29-s + 1.25·31-s − 0.883·32-s − 0.514·34-s − 0.676·35-s + 0.328·37-s + 0.648·38-s + 0.316·40-s + 0.468·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 13527 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13527 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(13527\) = \(3^{4} \cdot 167\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.862493\) |
| Root analytic conductor: | \(0.963693\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 13527,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.300722820\) |
| \(L(\frac12)\) | \(\approx\) | \(1.300722820\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 167 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) | ||
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.2.ab_c |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.5.c_g | |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.7.ac_g | |
| 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$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) | 2.13.af_be | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.19.ae_i | |
| 23 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) | 2.23.b_ab | |
| 29 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.29.ae_s | |
| 31 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.31.ah_bq | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.37.ac_ba | |
| 41 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.41.ad_ai | |
| 43 | $D_{4}$ | \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.43.h_ck | |
| 47 | $D_{4}$ | \( 1 + 11 T + 98 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.47.l_du | |
| 53 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.53.g_cw | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.ad_eo | |
| 61 | $D_{4}$ | \( 1 + 11 T + 76 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.61.l_cy | |
| 67 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.67.e_du | |
| 71 | $D_{4}$ | \( 1 + 5 T - 37 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.71.f_abl | |
| 73 | $D_{4}$ | \( 1 + 3 T - 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.73.d_am | |
| 79 | $D_{4}$ | \( 1 - 3 T - 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.79.ad_abb | |
| 83 | $D_{4}$ | \( 1 + 7 T + 23 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.83.h_x | |
| 89 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.89.e_ac | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.h_gi | |
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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
−16.1439412882, −15.5621419789, −15.3322311061, −14.6062119192, −14.1520722629, −13.8777537387, −13.3632544508, −12.9524752068, −12.1636909975, −11.7376286650, −11.3278426590, −11.0665017600, −10.1243028985, −9.55321105529, −8.83665588760, −8.46205154484, −7.94613074333, −7.22505904923, −6.42166508674, −5.98062688351, −4.83731212040, −4.55210456522, −3.87498366281, −3.23554567990, −1.49528059518, 1.49528059518, 3.23554567990, 3.87498366281, 4.55210456522, 4.83731212040, 5.98062688351, 6.42166508674, 7.22505904923, 7.94613074333, 8.46205154484, 8.83665588760, 9.55321105529, 10.1243028985, 11.0665017600, 11.3278426590, 11.7376286650, 12.1636909975, 12.9524752068, 13.3632544508, 13.8777537387, 14.1520722629, 14.6062119192, 15.3322311061, 15.5621419789, 16.1439412882