Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 6·5-s + 4·6-s − 4·7-s + 3·9-s + 12·10-s − 4·11-s − 4·12-s − 4·13-s + 8·14-s + 12·15-s − 4·16-s − 6·17-s − 6·18-s − 12·20-s + 8·21-s + 8·22-s + 17·25-s + 8·26-s − 4·27-s − 8·28-s + 4·29-s − 24·30-s + 8·32-s + 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 2.68·5-s + 1.63·6-s − 1.51·7-s + 9-s + 3.79·10-s − 1.20·11-s − 1.15·12-s − 1.10·13-s + 2.13·14-s + 3.09·15-s − 16-s − 1.45·17-s − 1.41·18-s − 2.68·20-s + 1.74·21-s + 1.70·22-s + 17/5·25-s + 1.56·26-s − 0.769·27-s − 1.51·28-s + 0.742·29-s − 4.38·30-s + 1.41·32-s + 1.39·33-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(12996\) = \(2^{2} \cdot 3^{2} \cdot 19^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.828636\) |
| Root analytic conductor: | \(0.954093\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\) |
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 | $C_2$ | \( 1 + p T + p T^{2} \) | |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) | 2.5.g_t |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.7.e_j | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.e_r | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.e_o | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.17.g_bn | |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.23.a_be | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.29.ae_bu | |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.31.a_ba | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.37.i_cw | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.41.i_de | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.43.ai_cz | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.47.i_dh | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.53.am_ew | |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) | 2.59.q_ha | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.61.ak_eh | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.67.ai_fe | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.71.q_hi | |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) | 2.73.w_kh | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.ai_be | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.83.ai_eo | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.ae_eo | |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | 2.97.u_li | |
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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
−16.5806200716, −16.2562118377, −15.8956818275, −15.6218536334, −15.3791583737, −14.7219277591, −13.4302811854, −13.2593691405, −12.4201439318, −12.1048029115, −11.7932373038, −11.0017900091, −10.8874895837, −10.0801905434, −9.91407057019, −8.83642738145, −8.56581084673, −7.64895552693, −7.47848778675, −6.89871092669, −6.41173993137, −5.13360434530, −4.51287023031, −3.81591641511, −2.74155671051, 0, 0, 2.74155671051, 3.81591641511, 4.51287023031, 5.13360434530, 6.41173993137, 6.89871092669, 7.47848778675, 7.64895552693, 8.56581084673, 8.83642738145, 9.91407057019, 10.0801905434, 10.8874895837, 11.0017900091, 11.7932373038, 12.1048029115, 12.4201439318, 13.2593691405, 13.4302811854, 14.7219277591, 15.3791583737, 15.6218536334, 15.8956818275, 16.2562118377, 16.5806200716