Dirichlet series
| L(s) = 1 | − 2-s − 4-s − 3·5-s + 3·8-s + 3·10-s − 3·13-s − 16-s + 17-s − 4·19-s + 3·20-s − 4·23-s + 25-s + 3·26-s − 7·29-s − 4·31-s − 5·32-s − 34-s − 11·37-s + 4·38-s − 9·40-s + 4·46-s + 4·47-s + 2·49-s − 50-s + 3·52-s + 7·58-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.34·5-s + 1.06·8-s + 0.948·10-s − 0.832·13-s − 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.670·20-s − 0.834·23-s + 1/5·25-s + 0.588·26-s − 1.29·29-s − 0.718·31-s − 0.883·32-s − 0.171·34-s − 1.80·37-s + 0.648·38-s − 1.42·40-s + 0.589·46-s + 0.583·47-s + 2/7·49-s − 0.141·50-s + 0.416·52-s + 0.919·58-s + 0.520·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(11664\) = \(2^{4} \cdot 3^{6}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.743706\) |
| Root analytic conductor: | \(0.928646\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 11664,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T + p T^{2} \) | |
| 3 | \( 1 \) | |||
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.5.d_i |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.7.a_ac | |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.11.a_g | |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.13.d_m | |
| 17 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.17.ab_au | |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.19.e_w | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.e_o | |
| 29 | $D_{4}$ | \( 1 + 7 T + 24 T^{2} + 7 p T^{3} + p^{2} T^{4} \) | 2.29.h_y | |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.31.e_ac | |
| 37 | $D_{4}$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.37.l_cq | |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.41.a_as | |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | 2.43.a_ak | |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.47.ae_be | |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.a_cs | |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.59.ae_abq | |
| 61 | $D_{4}$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.61.d_ae | |
| 67 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.67.am_fu | |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) | 2.71.a_bu | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.73.af_ds | |
| 79 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.79.am_fm | |
| 83 | $D_{4}$ | \( 1 + 8 T - 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.83.i_ak | |
| 89 | $D_{4}$ | \( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.89.aj_dw | |
| 97 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.97.ae_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
−16.7588071557, −16.1705636193, −15.7444392881, −15.2499725727, −14.7561806680, −14.2494325144, −13.7867616285, −13.0611363191, −12.6299798687, −12.0864548305, −11.6764821971, −10.9442538961, −10.5638578214, −9.94700982057, −9.36443412869, −8.81121960086, −8.24264513132, −7.70692271198, −7.37301604613, −6.62910051391, −5.53508367925, −4.92312069467, −3.96424687323, −3.70906390258, −2.06805291442, 0, 2.06805291442, 3.70906390258, 3.96424687323, 4.92312069467, 5.53508367925, 6.62910051391, 7.37301604613, 7.70692271198, 8.24264513132, 8.81121960086, 9.36443412869, 9.94700982057, 10.5638578214, 10.9442538961, 11.6764821971, 12.0864548305, 12.6299798687, 13.0611363191, 13.7867616285, 14.2494325144, 14.7561806680, 15.2499725727, 15.7444392881, 16.1705636193, 16.7588071557