Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s + 8-s + 2·9-s − 2·10-s − 5·11-s − 2·12-s − 5·13-s − 4·15-s − 3·16-s + 6·17-s − 2·18-s − 4·19-s + 2·20-s + 5·22-s + 2·23-s − 2·24-s − 2·25-s + 5·26-s − 6·27-s + 3·29-s + 4·30-s + 31-s + 5·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s + 0.353·8-s + 2/3·9-s − 0.632·10-s − 1.50·11-s − 0.577·12-s − 1.38·13-s − 1.03·15-s − 3/4·16-s + 1.45·17-s − 0.471·18-s − 0.917·19-s + 0.447·20-s + 1.06·22-s + 0.417·23-s − 0.408·24-s − 2/5·25-s + 0.980·26-s − 1.15·27-s + 0.557·29-s + 0.730·30-s + 0.179·31-s + 0.883·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1142\) = \(2 \cdot 571\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0728149\) |
| Root analytic conductor: | \(0.519463\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1142,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3374268099\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3374268099\) |
| \(L(\frac{3}{2})\) | 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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) | |
| 571 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 28 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_c |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.5.ac_g | |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) | 2.7.a_a | |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.11.f_s | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.f_u | |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.17.ag_bm | |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.19.e_s | |
| 23 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_s | |
| 29 | $D_{4}$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.29.ad_i | |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.31.ab_abe | |
| 37 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.37.b_m | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.41.c_bi | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.43.al_cw | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.47.ak_eg | |
| 53 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) | 2.53.a_au | |
| 59 | $D_{4}$ | \( 1 - 3 T + 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.59.ad_cg | |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.61.ab_aci | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.e_bm | |
| 71 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.71.e_u | |
| 73 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.73.o_fy | |
| 79 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.79.e_abq | |
| 83 | $D_{4}$ | \( 1 + T - 54 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.83.b_acc | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.89.g_fi | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) | 2.97.ax_mm | |
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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
−19.4481130748, −19.1499185592, −18.6184562244, −17.9660224271, −17.3897538631, −17.1295264330, −16.7509930181, −15.9723797342, −15.5568133604, −14.7190106662, −14.0100027502, −13.2886938320, −12.6891923372, −12.0899940840, −11.3705297274, −10.5822289444, −10.1797285795, −9.81415606453, −8.76426803796, −7.56213759903, −7.39937804289, −6.07712202493, −5.51875130209, −4.66444335517, −2.39675107461, 2.39675107461, 4.66444335517, 5.51875130209, 6.07712202493, 7.39937804289, 7.56213759903, 8.76426803796, 9.81415606453, 10.1797285795, 10.5822289444, 11.3705297274, 12.0899940840, 12.6891923372, 13.2886938320, 14.0100027502, 14.7190106662, 15.5568133604, 15.9723797342, 16.7509930181, 17.1295264330, 17.3897538631, 17.9660224271, 18.6184562244, 19.1499185592, 19.4481130748