Dirichlet series
| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s − 10-s − 7·11-s + 12-s − 5·13-s − 14-s − 15-s − 16-s + 3·17-s − 20-s − 21-s + 7·22-s − 3·23-s − 3·24-s − 4·25-s + 5·26-s + 4·27-s − 28-s − 29-s + 30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s − 0.316·10-s − 2.11·11-s + 0.288·12-s − 1.38·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.727·17-s − 0.223·20-s − 0.218·21-s + 1.49·22-s − 0.625·23-s − 0.612·24-s − 4/5·25-s + 0.980·26-s + 0.769·27-s − 0.188·28-s − 0.185·29-s + 0.182·30-s − 1.43·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 18960 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18960 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(18960\) = \(2^{4} \cdot 3 \cdot 5 \cdot 79\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.20890\) |
| Root analytic conductor: | \(1.04857\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 18960,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) | ||
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) | ||
| good | 7 | $D_{4}$ | \( 1 - T - 8 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.7.ab_ai |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.11.h_be | |
| 13 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.13.f_q | |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.17.ad_u | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.23.d_bs | |
| 29 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.29.b_bo | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.31.i_ck | |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) | 2.37.a_be | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.a_bu | |
| 43 | $D_{4}$ | \( 1 + 5 T + 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.43.f_e | |
| 47 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.47.c_w | |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) | 2.53.a_cc | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.c_ac | |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.61.a_ac | |
| 67 | $D_{4}$ | \( 1 + 2 T - 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.67.c_acc | |
| 71 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.71.ak_cs | |
| 73 | $D_{4}$ | \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.73.j_do | |
| 83 | $D_{4}$ | \( 1 + 21 T + 244 T^{2} + 21 p T^{3} + p^{2} T^{4} \) | 2.83.v_jk | |
| 89 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.89.am_di | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.97.ad_fk | |
| show more | ||||
| show less | ||||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.1128566977, −15.8414093530, −15.0219612938, −14.6804419851, −14.1091408514, −13.6393215802, −13.1331168778, −12.7113348070, −12.2108084190, −11.6630567649, −10.9216997499, −10.4302855445, −10.2286349301, −9.63361254899, −9.21062326269, −8.24627070675, −8.03571995443, −7.42839395310, −6.96410121702, −5.64815606763, −5.49790510997, −4.93103317386, −4.15672585530, −2.87981141912, −1.91471286515, 0, 1.91471286515, 2.87981141912, 4.15672585530, 4.93103317386, 5.49790510997, 5.64815606763, 6.96410121702, 7.42839395310, 8.03571995443, 8.24627070675, 9.21062326269, 9.63361254899, 10.2286349301, 10.4302855445, 10.9216997499, 11.6630567649, 12.2108084190, 12.7113348070, 13.1331168778, 13.6393215802, 14.1091408514, 14.6804419851, 15.0219612938, 15.8414093530, 16.1128566977