Dirichlet series
| L(s) = 1 | − 3·3-s − 2·5-s − 2·7-s + 4·9-s − 5·11-s − 4·13-s + 6·15-s − 5·17-s − 19-s + 6·21-s − 4·23-s + 2·25-s − 6·27-s − 2·29-s − 14·31-s + 15·33-s + 4·35-s + 5·37-s + 12·39-s − 41-s − 43-s − 8·45-s + 10·47-s + 15·51-s + 4·53-s + 10·55-s + 3·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s − 0.755·7-s + 4/3·9-s − 1.50·11-s − 1.10·13-s + 1.54·15-s − 1.21·17-s − 0.229·19-s + 1.30·21-s − 0.834·23-s + 2/5·25-s − 1.15·27-s − 0.371·29-s − 2.51·31-s + 2.61·33-s + 0.676·35-s + 0.821·37-s + 1.92·39-s − 0.156·41-s − 0.152·43-s − 1.19·45-s + 1.45·47-s + 2.10·51-s + 0.549·53-s + 1.34·55-s + 0.397·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(85280\) = \(2^{5} \cdot 5 \cdot 13 \cdot 41\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.43752\) |
| Root analytic conductor: | \(1.52703\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 85280,\ (\ :1/2, 1/2),\ 1)\) |
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 | 2 | \( 1 \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 3 T + p T^{2} ) \) | ||
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) | 2.3.d_f |
| 7 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_e | |
| 11 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.11.f_bb | |
| 17 | $D_{4}$ | \( 1 + 5 T + 29 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.17.f_bd | |
| 19 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.19.b_j | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.e_o | |
| 29 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_g | |
| 31 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) | 2.31.o_ee | |
| 37 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.37.af_bk | |
| 43 | $D_{4}$ | \( 1 + T - 26 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.43.b_aba | |
| 47 | $D_{4}$ | \( 1 - 10 T + 96 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.47.ak_ds | |
| 53 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.53.ae_g | |
| 59 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) | 2.59.a_cu | |
| 61 | $D_{4}$ | \( 1 - 5 T - 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.61.af_ap | |
| 67 | $D_{4}$ | \( 1 + 6 T + 36 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.67.g_bk | |
| 71 | $D_{4}$ | \( 1 + 5 T - 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.71.f_abh | |
| 73 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.73.ab_bt | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.79.o_gc | |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) | 2.83.a_fe | |
| 89 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.89.i_eo | |
| 97 | $D_{4}$ | \( 1 + 10 T + 104 T^{2} + 10 p T^{3} + p^{2} T^{4} \) | 2.97.k_ea | |
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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
−14.8674203095, −14.2906763893, −13.4336751911, −13.1961430689, −12.7768597857, −12.3722519320, −11.9820224469, −11.5439982771, −11.0225957597, −10.8106360816, −10.3987409333, −9.76095717204, −9.37035406688, −8.70626062509, −8.03148042778, −7.47177974025, −7.17214766692, −6.65166471530, −5.87394820759, −5.54377908567, −5.15379699824, −4.29881881046, −3.97656557291, −2.89306628083, −2.11814460777, 0, 0, 2.11814460777, 2.89306628083, 3.97656557291, 4.29881881046, 5.15379699824, 5.54377908567, 5.87394820759, 6.65166471530, 7.17214766692, 7.47177974025, 8.03148042778, 8.70626062509, 9.37035406688, 9.76095717204, 10.3987409333, 10.8106360816, 11.0225957597, 11.5439982771, 11.9820224469, 12.3722519320, 12.7768597857, 13.1961430689, 13.4336751911, 14.2906763893, 14.8674203095