Dirichlet series
| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·9-s − 2·10-s − 2·11-s − 5·13-s − 14-s + 16-s − 3·17-s − 2·18-s − 2·19-s − 2·20-s − 2·22-s − 2·25-s − 5·26-s + 3·27-s − 28-s − 8·31-s + 32-s − 3·34-s + 2·35-s − 2·36-s + 5·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.603·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.458·19-s − 0.447·20-s − 0.426·22-s − 2/5·25-s − 0.980·26-s + 0.577·27-s − 0.188·28-s − 1.43·31-s + 0.176·32-s − 0.514·34-s + 0.338·35-s − 1/3·36-s + 0.821·37-s − 0.324·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 49920 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49920 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(49920\) = \(2^{8} \cdot 3 \cdot 5 \cdot 13\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.18294\) |
| Root analytic conductor: | \(1.33569\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 49920,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 - T \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) | ||
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) | ||
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_g |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.c_ac | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.17.d_bi | |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.19.c_bi | |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.23.a_ac | |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.a_w | |
| 31 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.31.i_bu | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.37.af_y | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.ag_k | |
| 43 | $D_{4}$ | \( 1 + 3 T - 42 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.43.d_abq | |
| 47 | $D_{4}$ | \( 1 - 5 T + 84 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.47.af_dg | |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) | 2.53.a_acw | |
| 59 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.59.m_dy | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.61.am_es | |
| 67 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.67.ac_ck | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.71.l_fa | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.73.c_du | |
| 79 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.79.ac_dy | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.83.g_aby | |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) | 2.89.s_ig | |
| 97 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.97.e_abq | |
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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
−14.8861684015, −14.5556535322, −14.1881690993, −13.6215531367, −12.9929877677, −12.7200189109, −12.3967971588, −11.7263409887, −11.4379462312, −10.9743242921, −10.4779741596, −9.89298043147, −9.36818798180, −8.75516352868, −8.21802766651, −7.57924162352, −7.32091296919, −6.67625624062, −5.97202486880, −5.49362267309, −4.71813637803, −4.33178335489, −3.52874358548, −2.80832453442, −2.17216022900, 0, 2.17216022900, 2.80832453442, 3.52874358548, 4.33178335489, 4.71813637803, 5.49362267309, 5.97202486880, 6.67625624062, 7.32091296919, 7.57924162352, 8.21802766651, 8.75516352868, 9.36818798180, 9.89298043147, 10.4779741596, 10.9743242921, 11.4379462312, 11.7263409887, 12.3967971588, 12.7200189109, 12.9929877677, 13.6215531367, 14.1881690993, 14.5556535322, 14.8861684015