Dirichlet series
| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 4·7-s − 8-s − 2·9-s + 4·10-s + 4·11-s − 12-s − 8·13-s + 4·14-s + 4·15-s + 16-s − 2·17-s + 2·18-s + 8·19-s − 4·20-s + 4·21-s − 4·22-s + 24-s + 6·25-s + 8·26-s + 2·27-s − 4·28-s − 12·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s − 2/3·9-s + 1.26·10-s + 1.20·11-s − 0.288·12-s − 2.21·13-s + 1.06·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.471·18-s + 1.83·19-s − 0.894·20-s + 0.872·21-s − 0.852·22-s + 0.204·24-s + 6/5·25-s + 1.56·26-s + 0.384·27-s − 0.755·28-s − 2.22·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 110976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(110976\) = \(2^{7} \cdot 3 \cdot 17^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.07592\) |
| Root analytic conductor: | \(1.63096\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 110976,\ (\ :1/2, 1/2),\ 1)\) |
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$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| 17 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.7.e_s | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ae_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.i_bm | |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | 2.19.ai_cc | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.a_k | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.29.m_dm | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.31.e_by | |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.37.i_dm | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.41.m_dy | |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.43.i_dy | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.47.i_bu | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.53.ae_dq | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.59.aq_gk | |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.61.i_fi | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.i_di | |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.71.a_ec | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.73.e_fe | |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) | 2.79.au_jy | |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) | 2.83.y_ly | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.89.m_hq | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.97.e_fe | |
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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
−14.6871934666, −14.0380267054, −13.4366341279, −12.8599601742, −12.4406298356, −12.0011129381, −11.7067450165, −11.3267446972, −11.3184059703, −10.1363667552, −9.95744440217, −9.59406257733, −8.95116862257, −8.71110335004, −7.83292879902, −7.50224155159, −6.99640780174, −6.84041393535, −6.09068663105, −5.28300507382, −5.00716852165, −3.95839388323, −3.34589616522, −3.20324136736, −1.88616587730, 0, 0, 1.88616587730, 3.20324136736, 3.34589616522, 3.95839388323, 5.00716852165, 5.28300507382, 6.09068663105, 6.84041393535, 6.99640780174, 7.50224155159, 7.83292879902, 8.71110335004, 8.95116862257, 9.59406257733, 9.95744440217, 10.1363667552, 11.3184059703, 11.3267446972, 11.7067450165, 12.0011129381, 12.4406298356, 12.8599601742, 13.4366341279, 14.0380267054, 14.6871934666