Dirichlet series
| L(s) = 1 | − 3-s − 5-s − 2·9-s + 8·13-s + 15-s + 8·17-s − 4·19-s + 8·23-s − 2·25-s + 2·27-s − 12·29-s + 4·31-s + 8·37-s − 8·39-s + 12·41-s − 4·43-s + 2·45-s + 8·47-s − 14·49-s − 8·51-s + 12·53-s + 4·57-s + 8·59-s − 12·61-s − 8·65-s − 12·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 2.21·13-s + 0.258·15-s + 1.94·17-s − 0.917·19-s + 1.66·23-s − 2/5·25-s + 0.384·27-s − 2.22·29-s + 0.718·31-s + 1.31·37-s − 1.28·39-s + 1.87·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s − 2·49-s − 1.12·51-s + 1.64·53-s + 0.529·57-s + 1.04·59-s − 1.53·61-s − 0.992·65-s − 1.46·67-s − 0.963·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 15360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(15360\) = \(2^{10} \cdot 3 \cdot 5\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.979366\) |
| Root analytic conductor: | \(0.994801\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 15360,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.8794806064\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8794806064\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) | ||
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.7.a_o |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.11.a_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.13.ai_bm | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.17.ai_bu | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.e_bm | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.23.ai_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.29.m_da | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.ae_ck | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.ai_cc | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.41.am_dy | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.43.e_di | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.ai_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.53.am_da | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.ai_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.m_fm | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.m_fe | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.71.ai_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.73.u_iw | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.79.m_gc | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.83.e_gk | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.89.e_bm | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.97.au_iw | |
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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
−16.0387420541, −15.7488207479, −15.0277756685, −14.5765256398, −14.4078514439, −13.2768755254, −13.2686564900, −12.7887803972, −11.7668014526, −11.7666126827, −10.9716317562, −10.9076921437, −10.1922765354, −9.32846754645, −8.95538623117, −8.30430859978, −7.77199473906, −7.22612323208, −6.22207440469, −5.87146418849, −5.45564722344, −4.33436868576, −3.67478222653, −2.93279961257, −1.21912253618, 1.21912253618, 2.93279961257, 3.67478222653, 4.33436868576, 5.45564722344, 5.87146418849, 6.22207440469, 7.22612323208, 7.77199473906, 8.30430859978, 8.95538623117, 9.32846754645, 10.1922765354, 10.9076921437, 10.9716317562, 11.7666126827, 11.7668014526, 12.7887803972, 13.2686564900, 13.2768755254, 14.4078514439, 14.5765256398, 15.0277756685, 15.7488207479, 16.0387420541