Dirichlet series
| L(s) = 1 | + 2·3-s − 2·5-s + 9-s + 6·11-s − 4·15-s − 2·17-s + 2·19-s + 4·23-s + 2·25-s − 4·27-s − 2·29-s + 4·31-s + 12·33-s + 4·37-s − 6·41-s + 6·43-s − 2·45-s − 8·47-s − 6·49-s − 4·51-s − 2·53-s − 12·55-s + 4·57-s − 18·59-s + 4·61-s + 10·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.80·11-s − 1.03·15-s − 0.485·17-s + 0.458·19-s + 0.834·23-s + 2/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s + 2.08·33-s + 0.657·37-s − 0.937·41-s + 0.914·43-s − 0.298·45-s − 1.16·47-s − 6/7·49-s − 0.560·51-s − 0.274·53-s − 1.61·55-s + 0.529·57-s − 2.34·59-s + 0.512·61-s + 1.22·67-s + 0.963·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(18432\) = \(2^{11} \cdot 3^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.17524\) |
| Root analytic conductor: | \(1.04119\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 18432,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.479486074\) |
| \(L(\frac12)\) | \(\approx\) | \(1.479486074\) |
| \(L(\frac{3}{2})\) | 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 | $C_2$ | \( 1 - 2 T + p T^{2} \) | ||
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.c_c |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) | 2.7.a_g | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ag_w | |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.a_ak | |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.17.c_c | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.19.ac_ak | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.23.ae_o | |
| 29 | $D_{4}$ | \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.29.c_ao | |
| 31 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.31.ae_g | |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.37.ae_ac | |
| 41 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.41.g_by | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.43.ag_cs | |
| 47 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.47.i_da | |
| 53 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.53.c_bi | |
| 59 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) | 2.59.s_ha | |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.61.ae_bu | |
| 67 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.67.ak_cs | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.71.e_bu | |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.73.m_dy | |
| 79 | $D_{4}$ | \( 1 + 20 T + 230 T^{2} + 20 p T^{3} + p^{2} T^{4} \) | 2.79.u_iw | |
| 83 | $D_{4}$ | \( 1 - 2 T - 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.83.ac_adm | |
| 89 | $D_{4}$ | \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.89.c_ao | |
| 97 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.97.e_cs | |
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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
−15.6388431548, −15.3387029775, −14.7905923987, −14.4775533748, −14.0661810265, −13.6094598241, −12.9420941202, −12.5864804857, −11.8160826844, −11.3701186067, −11.3115530642, −10.2945362349, −9.65047505782, −9.18403429092, −8.80649502132, −8.27213573288, −7.70769473318, −7.13171606030, −6.56040333903, −5.87967844129, −4.73681815857, −4.20542641223, −3.46088107276, −2.91951872216, −1.56362959378, 1.56362959378, 2.91951872216, 3.46088107276, 4.20542641223, 4.73681815857, 5.87967844129, 6.56040333903, 7.13171606030, 7.70769473318, 8.27213573288, 8.80649502132, 9.18403429092, 9.65047505782, 10.2945362349, 11.3115530642, 11.3701186067, 11.8160826844, 12.5864804857, 12.9420941202, 13.6094598241, 14.0661810265, 14.4775533748, 14.7905923987, 15.3387029775, 15.6388431548