Dirichlet series
| L(s) = 1 | − 3-s − 2·9-s + 4·11-s − 8·13-s + 4·17-s − 4·19-s − 8·23-s − 6·25-s + 2·27-s + 16·29-s + 8·31-s − 4·33-s + 8·37-s + 8·39-s + 4·41-s + 4·43-s − 14·49-s − 4·51-s − 16·53-s + 4·57-s + 4·59-s + 8·61-s − 4·67-s + 8·69-s + 8·71-s + 4·73-s + 6·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 1.20·11-s − 2.21·13-s + 0.970·17-s − 0.917·19-s − 1.66·23-s − 6/5·25-s + 0.384·27-s + 2.97·29-s + 1.43·31-s − 0.696·33-s + 1.31·37-s + 1.28·39-s + 0.624·41-s + 0.609·43-s − 2·49-s − 0.560·51-s − 2.19·53-s + 0.529·57-s + 0.520·59-s + 1.02·61-s − 0.488·67-s + 0.963·69-s + 0.949·71-s + 0.468·73-s + 0.692·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(1536\) = \(2^{9} \cdot 3\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0979366\) |
| Root analytic conductor: | \(0.559417\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 1536,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4997926316\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4997926316\) |
| \(L(\frac{3}{2})\) | 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$ | \( ( 1 + T )( 1 + p T^{2} ) \) | ||
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.5.a_g |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.7.a_o | |
| 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 | |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.17.ae_bm | |
| 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$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.23.i_bu | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) | 2.29.aq_eo | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.31.ai_ck | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.37.ai_di | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.ae_w | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.43.ae_di | |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.47.a_dq | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) | 2.53.q_fe | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.ae_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.61.ai_dy | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.e_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$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.73.ae_di | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.79.i_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$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.89.ae_eo | |
| 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
−19.1792400873, −19.1551537949, −17.6707519500, −17.6322651559, −17.3311679255, −16.6272610009, −16.1152657336, −15.4732516475, −14.5925007839, −14.1819882616, −14.0930481386, −12.7248727791, −12.3172568607, −11.6664921937, −11.5820474614, −10.2342122988, −9.96467191573, −9.31735655616, −8.09899069409, −7.86574055999, −6.42897107073, −6.27282080647, −5.01452297596, −4.25303028693, −2.63895104553, 2.63895104553, 4.25303028693, 5.01452297596, 6.27282080647, 6.42897107073, 7.86574055999, 8.09899069409, 9.31735655616, 9.96467191573, 10.2342122988, 11.5820474614, 11.6664921937, 12.3172568607, 12.7248727791, 14.0930481386, 14.1819882616, 14.5925007839, 15.4732516475, 16.1152657336, 16.6272610009, 17.3311679255, 17.6322651559, 17.6707519500, 19.1551537949, 19.1792400873