Dirichlet series
| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 6·11-s + 2·13-s + 4·14-s + 16-s − 6·17-s + 4·19-s + 6·22-s − 25-s − 2·26-s − 4·28-s + 8·31-s − 32-s + 6·34-s + 8·37-s − 4·38-s − 2·43-s − 6·44-s + 18·47-s − 49-s + 50-s + 2·52-s + 12·53-s + 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.80·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 1.27·22-s − 1/5·25-s − 0.392·26-s − 0.755·28-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 1.31·37-s − 0.648·38-s − 0.304·43-s − 0.904·44-s + 2.62·47-s − 1/7·49-s + 0.141·50-s + 0.277·52-s + 1.64·53-s + 0.534·56-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(373248\) = \(2^{9} \cdot 3^{6}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(23.7986\) |
| Root analytic conductor: | \(2.20870\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 373248,\ (\ :1/2, 1/2),\ -1)\) |
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 | \( 1 \) | |||
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.a_b |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.7.e_r | |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) | 2.11.g_bf | |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.ac_c | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.g_bi | |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.19.ae_bq | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.a_k | |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.29.a_w | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.31.ai_cz | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.37.ai_di | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.a_bu | |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.43.c_g | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) | 2.47.as_gk | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.53.am_fd | |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.a_aba | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.61.ai_es | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) | 2.67.as_hi | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.71.g_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) | 2.73.s_ip | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.79.e_ck | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.83.ag_fj | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) | 2.89.s_gw | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 17 T + p T^{2} ) \) | 2.97.s_id | |
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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
−13.1830201225, −12.8571746215, −12.1853797856, −11.9118486001, −11.2680639273, −11.0333602201, −10.4854111065, −10.1152647782, −9.87204525884, −9.35322032957, −8.99827537344, −8.27873568760, −8.26804890939, −7.53175275772, −7.06148502534, −6.73860900675, −6.19336990210, −5.65487654062, −5.37363015538, −4.45990921767, −4.00726754984, −3.19909645813, −2.57462971648, −2.44850978333, −0.998052645277, 0, 0.998052645277, 2.44850978333, 2.57462971648, 3.19909645813, 4.00726754984, 4.45990921767, 5.37363015538, 5.65487654062, 6.19336990210, 6.73860900675, 7.06148502534, 7.53175275772, 8.26804890939, 8.27873568760, 8.99827537344, 9.35322032957, 9.87204525884, 10.1152647782, 10.4854111065, 11.0333602201, 11.2680639273, 11.9118486001, 12.1853797856, 12.8571746215, 13.1830201225